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Question

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

$\frac{1}{8}$
0
$\frac{1}{32}$
$\infty$

Hint:

In this question firstly we will solve the trigonometric equation and after eliminating the tan part. Then we will use the L'hospital theorem twice and later evaluate and put the value.

The correct answer is: $1 over 32$

$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.$

Related Questions to study

$Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction equals$

space space space Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction R a t o n a l i sin g space t h e space n u m e r a t o r comma equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction cross times fraction numerator square root of a plus 2 x end root plus square root of 3 x end root over denominator square root of a plus 2 x end root plus square root of 3 x end root end fraction cross times fraction numerator square root of 3 a plus x end root plus 2 square root of x over denominator square root of 3 a plus x end root plus 2 square root of x end fraction equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator left parenthesis a plus 2 x minus 3 x right parenthesis open parentheses square root of 3 a plus x end root plus 2 square root of x close parentheses over denominator left parenthesis 3 a plus x minus 4 x right parenthesis open parentheses square root of a plus 2 x end root plus square root of 3 x end root close parentheses end fraction equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of 3 a plus x end root plus 2 square root of x over denominator 3 open parentheses square root of a plus 2 x end root plus square root of 3 x end root close parentheses end fraction equals fraction numerator square root of 3 a plus a end root plus 2 square root of a over denominator 3 open parentheses square root of a plus 2 a end root plus square root of 3 a end root close parentheses end fraction equals 1 third cross times fraction numerator 4 square root of a over denominator 2 square root of 3 a end root end fraction equals fraction numerator 2 over denominator 3 square root of 3 end fraction

$Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction equals$

space space space Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction R a t o n a l i sin g space t h e space n u m e r a t o r comma equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction cross times fraction numerator square root of a plus 2 x end root plus square root of 3 x end root over denominator square root of a plus 2 x end root plus square root of 3 x end root end fraction cross times fraction numerator square root of 3 a plus x end root plus 2 square root of x over denominator square root of 3 a plus x end root plus 2 square root of x end fraction equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator left parenthesis a plus 2 x minus 3 x right parenthesis open parentheses square root of 3 a plus x end root plus 2 square root of x close parentheses over denominator left parenthesis 3 a plus x minus 4 x right parenthesis open parentheses square root of a plus 2 x end root plus square root of 3 x end root close parentheses end fraction equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of 3 a plus x end root plus 2 square root of x over denominator 3 open parentheses square root of a plus 2 x end root plus square root of 3 x end root close parentheses end fraction equals fraction numerator square root of 3 a plus a end root plus 2 square root of a over denominator 3 open parentheses square root of a plus 2 a end root plus square root of 3 a end root close parentheses end fraction equals 1 third cross times fraction numerator 4 square root of a over denominator 2 square root of 3 a end root end fraction equals fraction numerator 2 over denominator 3 square root of 3 end fraction

In $straight triangle A B C comma cot invisible function application A over 2 plus cot invisible function application B over 2 plus cot invisible function application C over 2 equals$

space space space c o t space A over 2 plus c o t space B over 2 plus c o t space C over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end fraction end root equals square root of fraction numerator s over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root left curly bracket left parenthesis s minus a right parenthesis plus left parenthesis s minus b right parenthesis plus left parenthesis s minus c right parenthesis right curly bracket m u l t i p l y i n g space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space w i t h space square root of s equals fraction numerator s over denominator square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root end fraction left curly bracket 3 s minus left parenthesis a plus b plus c right parenthesis right curly bracket equals fraction numerator s left parenthesis 3 s minus left parenthesis a plus b plus c right parenthesis right parenthesis over denominator increment end fraction equals fraction numerator begin display style fraction numerator left parenthesis a plus b plus c right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style minus begin display style fraction numerator 2 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style close square brackets over denominator increment end fraction equals fraction numerator open parentheses a plus b plus c close parentheses squared over denominator 4 increment end fraction

In $straight triangle A B C comma cot invisible function application A over 2 plus cot invisible function application B over 2 plus cot invisible function application C over 2 equals$

space space space c o t space A over 2 plus c o t space B over 2 plus c o t space C over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end fraction end root equals square root of fraction numerator s over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root left curly bracket left parenthesis s minus a right parenthesis plus left parenthesis s minus b right parenthesis plus left parenthesis s minus c right parenthesis right curly bracket m u l t i p l y i n g space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space w i t h space square root of s equals fraction numerator s over denominator square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root end fraction left curly bracket 3 s minus left parenthesis a plus b plus c right parenthesis right curly bracket equals fraction numerator s left parenthesis 3 s minus left parenthesis a plus b plus c right parenthesis right parenthesis over denominator increment end fraction equals fraction numerator begin display style fraction numerator left parenthesis a plus b plus c right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style minus begin display style fraction numerator 2 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style close square brackets over denominator increment end fraction equals fraction numerator open parentheses a plus b plus c close parentheses squared over denominator 4 increment end fraction

$Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals$

Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator 2 x squared plus x minus 3 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 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x squared plus x minus 3 end fraction equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 3 square root of x minus begin display style fraction numerator 3 over denominator 2 square root of x end fraction end style minus 2 over denominator 4 x plus 1 end fraction equals fraction numerator 3 cross times 1 space minus begin display style 3 over 2 end style cross times 1 minus 2 over denominator 4 cross times 1 space plus 1 end fraction equals fraction numerator 3 minus begin display style 3 over 2 end style minus 2 over denominator 5 end fraction equals fraction numerator begin display style fraction numerator 6 minus 3 minus 4 over denominator 2 end fraction end style over denominator 5 end fraction equals fraction numerator negative 1 over denominator 10 end fraction

$Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals$

Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator 2 x squared plus x minus 3 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 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x squared plus x minus 3 end fraction equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 3 square root of x minus begin display style fraction numerator 3 over denominator 2 square root of x end fraction end style minus 2 over denominator 4 x plus 1 end fraction equals fraction numerator 3 cross times 1 space minus begin display style 3 over 2 end style cross times 1 minus 2 over denominator 4 cross times 1 space plus 1 end fraction equals fraction numerator 3 minus begin display style 3 over 2 end style minus 2 over denominator 5 end fraction equals fraction numerator begin display style fraction numerator 6 minus 3 minus 4 over denominator 2 end fraction end style over denominator 5 end fraction equals fraction numerator negative 1 over denominator 10 end fraction

parallel

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 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 straight infinity end subscript fraction numerator begin display style fraction numerator d over denominator d x end fraction end style 11 x cubed minus 3 x plus 4 over denominator begin display style fraction numerator d over denominator d x end fraction end style 13 x cubed minus 5 x squared minus 7 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 33 x squared minus 3 over denominator 39 x squared minus 10 x end fraction A g a i n comma space 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 straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 33 x squared minus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 39 x squared minus 10 x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 66 x over denominator 78 x minus 10 end fraction A g a i n comma space 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 straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 66 x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 78 x minus 10 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript 66 over 78 equals 66 over 78 equals 11 over 13

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 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 straight infinity end subscript fraction numerator begin display style fraction numerator d over denominator d x end fraction end style 11 x cubed minus 3 x plus 4 over denominator begin display style fraction numerator d over denominator d x end fraction end style 13 x cubed minus 5 x squared minus 7 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 33 x squared minus 3 over denominator 39 x squared minus 10 x end fraction A g a i n comma space 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 straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 33 x squared minus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 39 x squared minus 10 x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 66 x over denominator 78 x minus 10 end fraction A g a i n comma space 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 straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 66 x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 78 x minus 10 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript 66 over 78 equals 66 over 78 equals 11 over 13

In $straight capital delta A B C$ , cot $A plus cot invisible function application B plus cot invisible function application C equals$

space space space space c o t space A plus c o t space B plus space c o t space C space equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

In $straight capital delta A B C$ , cot $A plus cot invisible function application B plus cot invisible function application C equals$

space space space space c o t space A plus c o t space B plus space c o t space C space equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction

open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x space space space comma space x greater or equal than 0 end cell row cell negative x comma space x less than 0 end cell end table close t a k i n g space x space p o s i t i v e comma limit as x rightwards arrow infinity of fraction numerator 8 x plus 3 x over denominator 3 x minus 2 x end fraction equals limit as x rightwards arrow infinity of fraction numerator 11 x over denominator x end fraction equals 11

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction

open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x space space space comma space x greater or equal than 0 end cell row cell negative x comma space x less than 0 end cell end table close t a k i n g space x space p o s i t i v e comma limit as x rightwards arrow infinity of fraction numerator 8 x plus 3 x over denominator 3 x minus 2 x end fraction equals limit as x rightwards arrow infinity of fraction numerator 11 x over denominator x end fraction equals 11

parallel

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction U sin g space L space H o s p t a l space t h e o r e m comma space equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 plus cos squared space x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x plus 2007 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 space cos space x space sin space x over denominator 1 end fraction equals 2

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction U sin g space L space H o s p t a l space t h e o r e m comma space equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 plus cos squared space x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x plus 2007 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 space cos space x space sin space x over denominator 1 end fraction equals 2

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses$

space space space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses equals space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses cross times fraction numerator square root of x squared plus x end root plus x over denominator square root of x squared plus x end root plus x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus x minus x squared over denominator square root of x squared plus x end root plus x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x over denominator square root of x squared plus x end root plus x end fraction D i v i d i n g space b y space x comma space equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of 1 plus begin display style 1 over x end style end root plus 1 end fraction equals fraction numerator 1 over denominator square root of 1 plus 1 end fraction equals 1 half.

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses$

space space space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses equals space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses cross times fraction numerator square root of x squared plus x end root plus x over denominator square root of x squared plus x end root plus x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus x minus x squared over denominator square root of x squared plus x end root plus x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x over denominator square root of x squared plus x end root plus x end fraction D i v i d i n g space b y space x comma space equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of 1 plus begin display style 1 over x end style end root plus 1 end fraction equals fraction numerator 1 over denominator square root of 1 plus 1 end fraction equals 1 half.

Elastic potential and gravitational potential energies are equal all the time.

The main difference between gravitational potential energy and elastic potential energy is that the origin of gravitational potential energy is the gravitational forces acting between two massive bodies whereas the origin of elastic potential energy is the electrostatic forces between molecules that make up a material
An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position

Elastic potential and gravitational potential energies are equal all the time.

The main difference between gravitational potential energy and elastic potential energy is that the origin of gravitational potential energy is the gravitational forces acting between two massive bodies whereas the origin of elastic potential energy is the electrostatic forces between molecules that make up a material
An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position

parallel

In $straight triangle A B C comma$ cot $A plus cot invisible function application B plus cot invisible function application C equals$

space space space space c o t space A plus c o t space B plus space c o t space C space equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

In $straight triangle A B C comma$ cot $A plus cot invisible function application B plus cot invisible function application C equals$

space space space space c o t space A plus c o t space B plus space c o t space C space equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

$text Lt end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis$

text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis equals text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis cross times fraction numerator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction equals straight c fraction numerator x plus 1 minus x over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction text = Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of x open parentheses square root of 1 plus begin display style 1 over x end style end root text +1 end text close parentheses end fraction equals space 1 over infinity cross times fraction numerator 1 over denominator square root of 1 plus begin display style 1 over infinity end style end root plus 1 end fraction equals 0 cross times 1 over 1 equals 0.

$text Lt end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis$

text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis equals text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis cross times fraction numerator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction equals straight c fraction numerator x plus 1 minus x over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction text = Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of x open parentheses square root of 1 plus begin display style 1 over x end style end root text +1 end text close parentheses end fraction equals space 1 over infinity cross times fraction numerator 1 over denominator square root of 1 plus begin display style 1 over infinity end style end root plus 1 end fraction equals 0 cross times 1 over 1 equals 0.

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator a subscript 0 plus a subscript 1 x to the power of 1 plus a subscript 2 x squared plus horizontal ellipsis plus a subscript n x to the power of n over denominator b subscript 0 plus b subscript 1 x to the power of 1 plus b subscript 2 x squared plus midline horizontal ellipsis plus b subscript m x to the power of w end fraction$ Where $a subscript n greater than 0 comma b subscript m greater than 0$ and $n greater than m$

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator a subscript 0 plus a subscript 1 x to the power of 1 plus a subscript 2 x squared plus horizontal ellipsis plus a subscript n x to the power of n over denominator b subscript 0 plus b subscript 1 x to the power of 1 plus b subscript 2 x squared plus midline horizontal ellipsis plus b subscript m x to the power of w end fraction$ Where $a subscript n greater than 0 comma b subscript m greater than 0$ and $n greater than m$

parallel

A ray of light is incident on an equilateral glass prism placed on a horizontal table For minimum deviation which of the following is true?

A ray of light is incident on an equilateral glass prism placed on a horizontal table For minimum deviation which of the following is true?

physics-General

Two identical thin isosceles prisms of refracting angle ‘A’ and refractive index m are placed with their bases touching each other Two parallel rays of light are incident on this system as shown The distance of the point where the rays converge from the prism is

Two identical thin isosceles prisms of refracting angle ‘A’ and refractive index m are placed with their bases touching each other Two parallel rays of light are incident on this system as shown The distance of the point where the rays converge from the prism is

physics-General

A ray of light strikes a plane mirror at an angle of incidence 45⁰ as shown in the figure After reflection, the ray passes through a prism of refractive index 1.50, whose apex angle is 4⁰ The angle through which the mirror should be rotated if the total deviation of the ray is to be 90⁰ is

A ray of light strikes a plane mirror at an angle of incidence 45⁰ as shown in the figure After reflection, the ray passes through a prism of refractive index 1.50, whose apex angle is 4⁰ The angle through which the mirror should be rotated if the total deviation of the ray is to be 90⁰ is

physics-General

parallel

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