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    Maths-
    General
    Easy

    Question

    L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction Is equal to

    1. pi
    2. negative pi
    3. pi over 2
    4. 1

    Hint:

    In the question we will simplify the equation using trigonometric identities.

    The correct answer is: pi

      L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction equals L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses straight pi open parentheses 1 minus sin squared straight x close parentheses close parentheses over denominator x squared end fraction equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses straight pi minus πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space open square brackets sin left parenthesis straight pi minus straight a right parenthesis equals sina close square brackets equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space space space space equals straight pi cross times 1 cross times 1 equals straight pi

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      General
      Maths-

      L t subscript x plus 0 end subscript fraction numerator left parenthesis 1 minus cos space 2 x right parenthesis left parenthesis 3 plus cos space x right parenthesis over denominator x tan space 4 x end fraction is equal to

      L t subscript x rightwards arrow 0 end subscript fraction numerator left parenthesis 1 minus cos space 2 x right parenthesis left parenthesis 3 plus cos space x right parenthesis over denominator x tan space 4 x end fraction equals L t subscript x rightwards arrow 0 end subscript space fraction numerator open parentheses 1 minus 1 plus 2 sin squared x close parentheses open parentheses 3 plus cos x close parentheses over denominator x.4 x end fraction fraction numerator 4 x over denominator tan 4 x end fraction space space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets equals L t subscript x rightwards arrow 0 end subscript fraction numerator 2 sin squared x over denominator 4 x squared end fraction space L t subscript x rightwards arrow 0 end subscript open parentheses 3 plus cos x close parentheses L t subscript x rightwards arrow 0 end subscript fraction numerator 4 x over denominator tan 4 x end fraction equals 1 half L t subscript x rightwards arrow 0 end subscript fraction numerator sin squared x over denominator x squared end fraction space L t subscript x rightwards arrow 0 end subscript open parentheses 3 plus cos x close parentheses L t subscript x rightwards arrow 0 end subscript fraction numerator 4 x over denominator tan 4 x end fraction equals 1 half open parentheses 3 plus 1 close parentheses 1 equals 4 over 2 equals 2

      L t subscript x plus 0 end subscript fraction numerator left parenthesis 1 minus cos space 2 x right parenthesis left parenthesis 3 plus cos space x right parenthesis over denominator x tan space 4 x end fraction is equal to

      Maths-General
      L t subscript x rightwards arrow 0 end subscript fraction numerator left parenthesis 1 minus cos space 2 x right parenthesis left parenthesis 3 plus cos space x right parenthesis over denominator x tan space 4 x end fraction equals L t subscript x rightwards arrow 0 end subscript space fraction numerator open parentheses 1 minus 1 plus 2 sin squared x close parentheses open parentheses 3 plus cos x close parentheses over denominator x.4 x end fraction fraction numerator 4 x over denominator tan 4 x end fraction space space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets equals L t subscript x rightwards arrow 0 end subscript fraction numerator 2 sin squared x over denominator 4 x squared end fraction space L t subscript x rightwards arrow 0 end subscript open parentheses 3 plus cos x close parentheses L t subscript x rightwards arrow 0 end subscript fraction numerator 4 x over denominator tan 4 x end fraction equals 1 half L t subscript x rightwards arrow 0 end subscript fraction numerator sin squared x over denominator x squared end fraction space L t subscript x rightwards arrow 0 end subscript open parentheses 3 plus cos x close parentheses L t subscript x rightwards arrow 0 end subscript fraction numerator 4 x over denominator tan 4 x end fraction equals 1 half open parentheses 3 plus 1 close parentheses 1 equals 4 over 2 equals 2
      General
      Maths-

      Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan begin display style space end style x minus sin space x over denominator x squared end fraction equals

      Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan begin display style space end style x minus sin space x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x over denominator cos x end fraction end style minus sin x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x minus sin x space cos x over denominator cos x end fraction end style over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator sin x left parenthesis 1 minus cos x right parenthesis over denominator x squared. cos x end fraction equals limit as x rightwards arrow 0 of fraction numerator sin x space over denominator x end fraction fraction numerator x over denominator cos x end fraction fraction numerator 1 minus cos x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction limit as x rightwards arrow 0 of fraction numerator x over denominator cos space x end fraction limit as x rightwards arrow 0 of fraction numerator 2 sin squared begin display style x over 2 end style over denominator open parentheses begin display style x over 2 end style close parentheses squared 4 end fraction space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets equals 1. fraction numerator 0 over denominator cos 0 end fraction.2 equals 0

      Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan begin display style space end style x minus sin space x over denominator x squared end fraction equals

      Maths-General
      Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan begin display style space end style x minus sin space x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x over denominator cos x end fraction end style minus sin x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x minus sin x space cos x over denominator cos x end fraction end style over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator sin x left parenthesis 1 minus cos x right parenthesis over denominator x squared. cos x end fraction equals limit as x rightwards arrow 0 of fraction numerator sin x space over denominator x end fraction fraction numerator x over denominator cos x end fraction fraction numerator 1 minus cos x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction limit as x rightwards arrow 0 of fraction numerator x over denominator cos space x end fraction limit as x rightwards arrow 0 of fraction numerator 2 sin squared begin display style x over 2 end style over denominator open parentheses begin display style x over 2 end style close parentheses squared 4 end fraction space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets equals 1. fraction numerator 0 over denominator cos 0 end fraction.2 equals 0
      General
      Maths-

      Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals

      Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction L e t space y equals square root of 1 plus x end root space space space space space space space space space space space space space space space space space space space space x rightwards arrow 8 rightwards double arrow y squared equals 1 plus x space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus x end root rightwards double arrow x equals y squared minus 1 space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus 8 end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow 3 Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 1 minus 8 end fraction equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 9 end fraction equals limit as y rightwards arrow 3 of fraction numerator open parentheses square root of 1 plus y end root minus 2 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator 1 plus y minus 4 over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator open parentheses y squared minus 3 squared close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator left parenthesis y minus 3 right parenthesis left parenthesis y plus 3 right parenthesis open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator 1 over denominator open parentheses y plus 3 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals fraction numerator 1 over denominator left parenthesis 3 plus 3 right parenthesis left parenthesis square root of 1 plus 3 end root plus 2 right parenthesis end fraction equals fraction numerator 1 over denominator 6 cross times 4 end fraction equals 1 over 24

      Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals

      Maths-General
      Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction L e t space y equals square root of 1 plus x end root space space space space space space space space space space space space space space space space space space space space x rightwards arrow 8 rightwards double arrow y squared equals 1 plus x space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus x end root rightwards double arrow x equals y squared minus 1 space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus 8 end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow 3 Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 1 minus 8 end fraction equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 9 end fraction equals limit as y rightwards arrow 3 of fraction numerator open parentheses square root of 1 plus y end root minus 2 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator 1 plus y minus 4 over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator open parentheses y squared minus 3 squared close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator left parenthesis y minus 3 right parenthesis left parenthesis y plus 3 right parenthesis open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator 1 over denominator open parentheses y plus 3 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals fraction numerator 1 over denominator left parenthesis 3 plus 3 right parenthesis left parenthesis square root of 1 plus 3 end root plus 2 right parenthesis end fraction equals fraction numerator 1 over denominator 6 cross times 4 end fraction equals 1 over 24
      parallel
      General
      Maths-

      L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction equals

      L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction u sin g space L h o s p i t a l space t h e o r e m comma space equals L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style square root of x plus 4 end root minus 2 end fraction equals L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator begin display style fraction numerator 1 over denominator 2 square root of x plus 4 end root end fraction end style end fraction equals L t subscript x not stretchy rightwards arrow 0 end subscript 2 square root of x plus 4 end root equals 2 square root of 0 plus 4 end root equals 2 cross times 2 equals 4

      L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction equals

      Maths-General
      L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction u sin g space L h o s p i t a l space t h e o r e m comma space equals L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style square root of x plus 4 end root minus 2 end fraction equals L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator begin display style fraction numerator 1 over denominator 2 square root of x plus 4 end root end fraction end style end fraction equals L t subscript x not stretchy rightwards arrow 0 end subscript 2 square root of x plus 4 end root equals 2 square root of 0 plus 4 end root equals 2 cross times 2 equals 4
      General
      Maths-

      Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction equals

      Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction u sin g space L space h o s p i t a l space t h e o r e m comma space equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 plus x end root minus begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 minus x end root over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style minus begin display style fraction numerator negative 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style over denominator 1 end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction equals fraction numerator 1 over denominator 3 open parentheses 1 plus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction equals 1 third plus 1 third equals 2 over 3

      Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction equals

      Maths-General
      Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction u sin g space L space h o s p i t a l space t h e o r e m comma space equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 plus x end root minus begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 minus x end root over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style minus begin display style fraction numerator negative 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style over denominator 1 end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction equals fraction numerator 1 over denominator 3 open parentheses 1 plus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction equals 1 third plus 1 third equals 2 over 3
      General
      Maths-

      If f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root comma text  then  end text L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction equals

      Given, f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root
      L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction u sin g space L space h o s p i t a l space t h e o r e m comma equals L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x minus 1 end fraction equals L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f apostrophe left parenthesis x right parenthesis over denominator 1 end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript f apostrophe left parenthesis x right parenthesis equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator d left parenthesis negative square root of 25 minus x squared end root right parenthesis over denominator d x end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator negative 1 cross times negative 2 over denominator 2 square root of 25 minus x squared end root end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator 1 over denominator square root of 25 minus x squared end root end fraction equals fraction numerator 1 over denominator square root of 25 minus 1 end root end fraction equals fraction numerator 1 over denominator square root of 24 end fraction

      If f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root comma text  then  end text L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction equals

      Maths-General
      Given, f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root
      L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction u sin g space L space h o s p i t a l space t h e o r e m comma equals L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x minus 1 end fraction equals L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f apostrophe left parenthesis x right parenthesis over denominator 1 end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript f apostrophe left parenthesis x right parenthesis equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator d left parenthesis negative square root of 25 minus x squared end root right parenthesis over denominator d x end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator negative 1 cross times negative 2 over denominator 2 square root of 25 minus x squared end root end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator 1 over denominator square root of 25 minus x squared end root end fraction equals fraction numerator 1 over denominator square root of 25 minus 1 end root end fraction equals fraction numerator 1 over denominator square root of 24 end fraction
      parallel
      General
      maths-

      L t subscript x not stretchy rightwards arrow 2 end subscript open parentheses fraction numerator square root of 1 minus cos space left curly bracket 2 left parenthesis x minus 2 right parenthesis factorial end root over denominator x minus 2 end fraction close parentheses equals

      L t subscript x not stretchy rightwards arrow 2 end subscript open parentheses fraction numerator square root of 1 minus cos space left curly bracket 2 left parenthesis x minus 2 right parenthesis factorial end root over denominator x minus 2 end fraction close parentheses equals

      maths-General
      General
      Maths-

      If L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared  then the values of a and b are


      space space space space L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared rightwards double arrow e to the power of limit as x rightwards arrow infinity of 2 x open parentheses 1 plus a over x plus b over x squared minus 1 close parentheses end exponent equals e to the power of 2 space space space space space space space space space end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets u sin g space f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals g left parenthesis x right parenthesis open square brackets f left parenthesis x right parenthesis minus 1 close square brackets close square brackets rightwards double arrow limit as x rightwards arrow infinity of 2 x open parentheses a over x plus b over x squared close parentheses equals 2 rightwards double arrow 2 limit as x rightwards arrow infinity of open parentheses a plus b over x close parentheses equals 2 rightwards double arrow 2. left parenthesis a plus 0 right parenthesis equals 2 rightwards double arrow a equals 1 space a n d space b element of R

      If L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared  then the values of a and b are


      Maths-General
      space space space space L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared rightwards double arrow e to the power of limit as x rightwards arrow infinity of 2 x open parentheses 1 plus a over x plus b over x squared minus 1 close parentheses end exponent equals e to the power of 2 space space space space space space space space space end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets u sin g space f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals g left parenthesis x right parenthesis open square brackets f left parenthesis x right parenthesis minus 1 close square brackets close square brackets rightwards double arrow limit as x rightwards arrow infinity of 2 x open parentheses a over x plus b over x squared close parentheses equals 2 rightwards double arrow 2 limit as x rightwards arrow infinity of open parentheses a plus b over x close parentheses equals 2 rightwards double arrow 2. left parenthesis a plus 0 right parenthesis equals 2 rightwards double arrow a equals 1 space a n d space b element of R
      General
      Maths-

      text  In  end text straight triangle A B C comma straight capital sigma fraction numerator a squared sin invisible function application left parenthesis B minus C right parenthesis over denominator sin invisible function application A end fraction equals

      text  In  end text straight triangle A B C comma straight capital sigma fraction numerator a squared sin degree left parenthesis B minus C right parenthesis over denominator sin degree A end fraction equals fraction numerator straight a squared sin degree left parenthesis straight B minus straight C right parenthesis over denominator sin degree straight A end fraction plus fraction numerator straight b squared sin degree left parenthesis straight C minus straight A right parenthesis over denominator sin degree straight B end fraction plus fraction numerator straight c squared sin degree left parenthesis straight A minus straight B right parenthesis over denominator sin degree straight C end fraction using space sine space formula comma fraction numerator straight a over denominator sin space straight A end fraction equals fraction numerator straight b over denominator sin space straight B end fraction equals fraction numerator straight c over denominator sin space straight C end fraction equals straight k fraction numerator straight a squared sin degree left parenthesis straight B minus straight C right parenthesis over denominator sin degree straight A end fraction plus fraction numerator straight b squared sin degree left parenthesis straight C minus straight A right parenthesis over denominator sin degree straight B end fraction plus fraction numerator straight c squared sin degree left parenthesis straight A minus straight B right parenthesis over denominator sin degree straight C end fraction equals fraction numerator straight k squared sin squared straight A open parentheses sin open parentheses straight B minus straight C close parentheses close parentheses over denominator sinA end fraction plus fraction numerator straight k squared sin squared straight B open parentheses sin open parentheses straight C minus straight A close parentheses close parentheses over denominator sin space straight B end fraction plus fraction numerator straight k squared sin squared straight C open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses over denominator sinC end fraction equals straight k squared open square brackets sin space straight A space sin open parentheses straight B minus straight C close parentheses plus sinB left parenthesis sin open parentheses straight C minus straight A close parentheses plus sinC open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin open parentheses straight pi minus open parentheses straight B plus straight C close parentheses close parentheses space sin open parentheses straight B minus straight C close parentheses plus sin open parentheses straight pi minus open parentheses straight C plus straight A close parentheses close parentheses left parenthesis sin open parentheses straight C minus straight A close parentheses plus sin open parentheses straight pi minus open parentheses straight A plus straight B close parentheses close parentheses open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin left parenthesis straight B plus straight C right parenthesis space sin open parentheses straight B minus straight C close parentheses plus sin left parenthesis straight C plus straight A right parenthesis left parenthesis sin open parentheses straight C minus straight A close parentheses plus sin left parenthesis straight A plus straight B right parenthesis open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin squared straight B minus sin squared straight C plus sin squared straight C minus sin squared straight A plus sin squared straight A minus sin squared straight B close square brackets equals straight k squared cross times 0 equals 0.

      text  In  end text straight triangle A B C comma straight capital sigma fraction numerator a squared sin invisible function application left parenthesis B minus C right parenthesis over denominator sin invisible function application A end fraction equals

      Maths-General
      text  In  end text straight triangle A B C comma straight capital sigma fraction numerator a squared sin degree left parenthesis B minus C right parenthesis over denominator sin degree A end fraction equals fraction numerator straight a squared sin degree left parenthesis straight B minus straight C right parenthesis over denominator sin degree straight A end fraction plus fraction numerator straight b squared sin degree left parenthesis straight C minus straight A right parenthesis over denominator sin degree straight B end fraction plus fraction numerator straight c squared sin degree left parenthesis straight A minus straight B right parenthesis over denominator sin degree straight C end fraction using space sine space formula comma fraction numerator straight a over denominator sin space straight A end fraction equals fraction numerator straight b over denominator sin space straight B end fraction equals fraction numerator straight c over denominator sin space straight C end fraction equals straight k fraction numerator straight a squared sin degree left parenthesis straight B minus straight C right parenthesis over denominator sin degree straight A end fraction plus fraction numerator straight b squared sin degree left parenthesis straight C minus straight A right parenthesis over denominator sin degree straight B end fraction plus fraction numerator straight c squared sin degree left parenthesis straight A minus straight B right parenthesis over denominator sin degree straight C end fraction equals fraction numerator straight k squared sin squared straight A open parentheses sin open parentheses straight B minus straight C close parentheses close parentheses over denominator sinA end fraction plus fraction numerator straight k squared sin squared straight B open parentheses sin open parentheses straight C minus straight A close parentheses close parentheses over denominator sin space straight B end fraction plus fraction numerator straight k squared sin squared straight C open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses over denominator sinC end fraction equals straight k squared open square brackets sin space straight A space sin open parentheses straight B minus straight C close parentheses plus sinB left parenthesis sin open parentheses straight C minus straight A close parentheses plus sinC open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin open parentheses straight pi minus open parentheses straight B plus straight C close parentheses close parentheses space sin open parentheses straight B minus straight C close parentheses plus sin open parentheses straight pi minus open parentheses straight C plus straight A close parentheses close parentheses left parenthesis sin open parentheses straight C minus straight A close parentheses plus sin open parentheses straight pi minus open parentheses straight A plus straight B close parentheses close parentheses open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin left parenthesis straight B plus straight C right parenthesis space sin open parentheses straight B minus straight C close parentheses plus sin left parenthesis straight C plus straight A right parenthesis left parenthesis sin open parentheses straight C minus straight A close parentheses plus sin left parenthesis straight A plus straight B right parenthesis open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin squared straight B minus sin squared straight C plus sin squared straight C minus sin squared straight A plus sin squared straight A minus sin squared straight B close square brackets equals straight k squared cross times 0 equals 0.
      parallel
      General
      Maths-

      If L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k then k=

      L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k
      L.H.S=L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction
      using L hospital theorem,
      L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d straight x end fraction end style log begin display style space end style left parenthesis 3 plus x right parenthesis minus begin display style fraction numerator d blank over denominator d x end fraction end style log space left parenthesis 3 minus x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 plus straight x end fraction end style plus begin display style fraction numerator 1 over denominator 3 minus straight x end fraction end style over denominator 1 end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 3 minus straight x plus 3 plus straight x over denominator left parenthesis 3 plus straight x right parenthesis left parenthesis 3 minus straight x right parenthesis end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 6 over denominator 9 plus straight x squared end fraction equals fraction numerator 6 over denominator 9 plus 0 end fraction equals 6 over 9 equals 2 over 3
      So, k=2 over 3

      If L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k then k=

      Maths-General
      L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k
      L.H.S=L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction
      using L hospital theorem,
      L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d straight x end fraction end style log begin display style space end style left parenthesis 3 plus x right parenthesis minus begin display style fraction numerator d blank over denominator d x end fraction end style log space left parenthesis 3 minus x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 plus straight x end fraction end style plus begin display style fraction numerator 1 over denominator 3 minus straight x end fraction end style over denominator 1 end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 3 minus straight x plus 3 plus straight x over denominator left parenthesis 3 plus straight x right parenthesis left parenthesis 3 minus straight x right parenthesis end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 6 over denominator 9 plus straight x squared end fraction equals fraction numerator 6 over denominator 9 plus 0 end fraction equals 6 over 9 equals 2 over 3
      So, k=2 over 3
      General
      maths-

      Let proportional to and beta be the distinct roots of  a x squared plus b x plus c equals 0 then fraction numerator L t over denominator x plus a end fraction fraction numerator 1 minus cos space open parentheses a x squared plus b x plus c close parentheses over denominator left parenthesis x minus alpha right parenthesis squared end fraction

      Let proportional to and beta be the distinct roots of  a x squared plus b x plus c equals 0 then fraction numerator L t over denominator x plus a end fraction fraction numerator 1 minus cos space open parentheses a x squared plus b x plus c close parentheses over denominator left parenthesis x minus alpha right parenthesis squared end fraction

      maths-General
      General
      Maths-

      In straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals

      In space straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis L e t space fraction numerator sin space A thin space over denominator a end fraction equals fraction numerator sin space B over denominator b end fraction equals fraction numerator sin space C over denominator c end fraction equals k cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction comma cos space A equals fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction comma cos space A equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction space space space space space space straight a cubed sin left parenthesis straight B minus straight C right parenthesis equals a cubed open parentheses fraction numerator k b left parenthesis a squared plus b squared minus c squared right parenthesis over denominator 2 a b end fraction minus fraction numerator k c left parenthesis c squared plus a squared minus b squared right parenthesis over denominator 2 a c end fraction close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses a squared plus b squared minus c squared minus c squared minus a squared plus b squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses 2 b squared minus 2 c squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction 2. open parentheses b squared minus c squared close parentheses equals k a squared open parentheses b squared minus c squared close parentheses s i m i l a r l y comma space space b cubed sin left parenthesis C minus A right parenthesis equals k b squared left parenthesis c squared minus a squared right parenthesis s i m i l a r l y comma space c cubed sin space left parenthesis A minus B right parenthesis equals k c squared left parenthesis a squared minus b squared right parenthesis N o w comma space sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis equals k a squared open parentheses b squared minus c squared close parentheses plus k b squared left parenthesis c squared minus a squared right parenthesis plus k c squared left parenthesis a squared minus b squared right parenthesis equals k a squared b squared minus k a squared c squared plus k b squared c squared minus k a squared b squared plus k a squared c squared minus k b squared c squared equals 0.

      In straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals

      Maths-General
      In space straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis L e t space fraction numerator sin space A thin space over denominator a end fraction equals fraction numerator sin space B over denominator b end fraction equals fraction numerator sin space C over denominator c end fraction equals k cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction comma cos space A equals fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction comma cos space A equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction space space space space space space straight a cubed sin left parenthesis straight B minus straight C right parenthesis equals a cubed open parentheses fraction numerator k b left parenthesis a squared plus b squared minus c squared right parenthesis over denominator 2 a b end fraction minus fraction numerator k c left parenthesis c squared plus a squared minus b squared right parenthesis over denominator 2 a c end fraction close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses a squared plus b squared minus c squared minus c squared minus a squared plus b squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses 2 b squared minus 2 c squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction 2. open parentheses b squared minus c squared close parentheses equals k a squared open parentheses b squared minus c squared close parentheses s i m i l a r l y comma space space b cubed sin left parenthesis C minus A right parenthesis equals k b squared left parenthesis c squared minus a squared right parenthesis s i m i l a r l y comma space c cubed sin space left parenthesis A minus B right parenthesis equals k c squared left parenthesis a squared minus b squared right parenthesis N o w comma space sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis equals k a squared open parentheses b squared minus c squared close parentheses plus k b squared left parenthesis c squared minus a squared right parenthesis plus k c squared left parenthesis a squared minus b squared right parenthesis equals k a squared b squared minus k a squared c squared plus k b squared c squared minus k a squared b squared plus k a squared c squared minus k b squared c squared equals 0.
      parallel
      General
      maths-

      Lt subscript x not stretchy rightwards arrow 1 end subscript space left parenthesis 1 minus x right parenthesis tan space open parentheses pi x over 2 close parentheses equals

      Lt subscript x not stretchy rightwards arrow 1 end subscript space left parenthesis 1 minus x right parenthesis tan space open parentheses pi x over 2 close parentheses equals

      maths-General
      General
      Maths-

      In straight capital delta A B C comma a squared cot invisible function application A plus b squared cot invisible function application B plus C squared cot invisible function application C equals

      increment equals fraction numerator a over denominator 2 sin space A end fraction equals fraction numerator b over denominator 2 space sin space b end fraction equals fraction numerator c over denominator 2 space sin space C end fraction a squared c o t space A plus b squared c o t space B plus c squared c o t space B equals left parenthesis 2 increment sin space A right parenthesis squared. c o t space A plus open parentheses 2 space increment sin space B close parentheses squared. c o t space B plus open parentheses 2 space increment sin space close parentheses squared. c o t space C equals 4 increment squared left parenthesis sin squared space A. c o t space A plus sin squared B. c o t space B plus sin squared C. c o t space C right parenthesis equals 2 increment squared left parenthesis 2 sin space A. cos space A plus 2 sin space B space. cos space B plus 2 sin space C. cos space C right parenthesis equals 2 increment squared left parenthesis sin space 2 A plus sin space 2 B plus sin space 2 C right parenthesis equals 2 increment squared left parenthesis 2 sin left parenthesis A plus B right parenthesis. cos left parenthesis A minus B right parenthesis plus 2. sin space C. cos space C right parenthesis equals 2 increment squared left parenthesis 2 sin left parenthesis straight pi minus straight c right parenthesis. cos left parenthesis A minus B right parenthesis plus 2. sin space C. cos space left parenthesis straight pi minus left parenthesis straight A plus straight B right parenthesis right parenthesis right parenthesis equals 4 increment squared sin space C left parenthesis cos left parenthesis A plus B right parenthesis plus cos left parenthesis A minus B right parenthesis right parenthesis equals 8 increment squared sin space A. sin space B. space sin space C equals 8 increment squared fraction numerator a over denominator 2 increment end fraction. fraction numerator b over denominator 2 increment end fraction. fraction numerator c over denominator 2 increment end fraction equals fraction numerator a b c over denominator increment end fraction

      In straight capital delta A B C comma a squared cot invisible function application A plus b squared cot invisible function application B plus C squared cot invisible function application C equals

      Maths-General
      increment equals fraction numerator a over denominator 2 sin space A end fraction equals fraction numerator b over denominator 2 space sin space b end fraction equals fraction numerator c over denominator 2 space sin space C end fraction a squared c o t space A plus b squared c o t space B plus c squared c o t space B equals left parenthesis 2 increment sin space A right parenthesis squared. c o t space A plus open parentheses 2 space increment sin space B close parentheses squared. c o t space B plus open parentheses 2 space increment sin space close parentheses squared. c o t space C equals 4 increment squared left parenthesis sin squared space A. c o t space A plus sin squared B. c o t space B plus sin squared C. c o t space C right parenthesis equals 2 increment squared left parenthesis 2 sin space A. cos space A plus 2 sin space B space. cos space B plus 2 sin space C. cos space C right parenthesis equals 2 increment squared left parenthesis sin space 2 A plus sin space 2 B plus sin space 2 C right parenthesis equals 2 increment squared left parenthesis 2 sin left parenthesis A plus B right parenthesis. cos left parenthesis A minus B right parenthesis plus 2. sin space C. cos space C right parenthesis equals 2 increment squared left parenthesis 2 sin left parenthesis straight pi minus straight c right parenthesis. cos left parenthesis A minus B right parenthesis plus 2. sin space C. cos space left parenthesis straight pi minus left parenthesis straight A plus straight B right parenthesis right parenthesis right parenthesis equals 4 increment squared sin space C left parenthesis cos left parenthesis A plus B right parenthesis plus cos left parenthesis A minus B right parenthesis right parenthesis equals 8 increment squared sin space A. sin space B. space sin space C equals 8 increment squared fraction numerator a over denominator 2 increment end fraction. fraction numerator b over denominator 2 increment end fraction. fraction numerator c over denominator 2 increment end fraction equals fraction numerator a b c over denominator increment end fraction
      General
      Maths-

      Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals

      space space space space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets tan begin display style straight pi over 4 end style minus tan space x over 2 close square brackets over denominator open square brackets tan space begin display style straight pi over 4 end style plus tan space x over 2 close square brackets end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space tan space open parentheses straight pi over 4 minus x over 2 close parentheses. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator tan space open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator 4 left square bracket pi minus 2 x right square bracket squared end fraction equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket squared end fraction U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative cos space x over denominator 2 left parenthesis straight pi minus 2 straight x right parenthesis. left parenthesis negative 2 right parenthesis end fraction equals 1 over 16 space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator left parenthesis straight pi minus 2 straight x right parenthesis end fraction equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative sin space x over denominator negative 2 end fraction equals 1 over 32 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space sin space x equals 1 over 32 cross times sin space straight pi over 2 equals 1 over 32 space cross times 1 equals 1 over 32.

      Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals

      Maths-General
      space space space space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets tan begin display style straight pi over 4 end style minus tan space x over 2 close square brackets over denominator open square brackets tan space begin display style straight pi over 4 end style plus tan space x over 2 close square brackets end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space tan space open parentheses straight pi over 4 minus x over 2 close parentheses. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator tan space open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator 4 left square bracket pi minus 2 x right square bracket squared end fraction equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket squared end fraction U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative cos space x over denominator 2 left parenthesis straight pi minus 2 straight x right parenthesis. left parenthesis negative 2 right parenthesis end fraction equals 1 over 16 space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator left parenthesis straight pi minus 2 straight x right parenthesis end fraction equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative sin space x over denominator negative 2 end fraction equals 1 over 32 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space sin space x equals 1 over 32 cross times sin space straight pi over 2 equals 1 over 32 space cross times 1 equals 1 over 32.
      parallel

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