Maths-
General
Easy

Question

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

  1. 1
  2. -1
  3. 0
  4. 1 half

Hint:

In this question first we will rationalize the equation then we will divide the equation by x and then put the value of x.

The correct answer is: 1 half


    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.

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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
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    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

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    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.
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d i v i d 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 b y space x comma
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a plus begin display style b over x end style over denominator open square brackets square root of 1 plus begin display style a over x end style plus begin display style b over x squared end style end root plus 1 close square brackets end fraction
equals fraction numerator a over denominator square root of 1 plus 1 end fraction
equals a over 2

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open square brackets square root of x squared plus a x plus b end root minus x close square brackets

    Maths-General
    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open square brackets square root of x squared plus a x plus b end root minus x close square brackets
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator open square brackets square root of x squared plus a x plus b end root minus x close square brackets open square brackets square root of x squared plus a x plus b end root plus x close square brackets over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus a x plus b minus x squared over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a x plus b over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
d i v i d 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 b y space x comma
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a plus begin display style b over x end style over denominator open square brackets square root of 1 plus begin display style a over x end style plus begin display style b over x squared end style end root plus 1 close square brackets end fraction
equals fraction numerator a over denominator square root of 1 plus 1 end fraction
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    L t subscript n not stretchy rightwards arrow straight infinity end subscript sum from n equals 1 to n of   open square brackets fraction numerator 1 over denominator left parenthesis 2 n plus 1 right parenthesis left parenthesis 2 n plus 3 right parenthesis end fraction close square brackets

    L t subscript n not stretchy rightwards arrow straight infinity end subscript sum from n equals 1 to n of   open square brackets fraction numerator 1 over denominator left parenthesis 2 n plus 1 right parenthesis left parenthesis 2 n plus 3 right parenthesis end fraction close square brackets
equals limit as n rightwards arrow infinity of fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus....... plus fraction numerator 1 over denominator open parentheses 2 n plus 1 close parentheses left parenthesis 2 n plus 3 right parenthesis end fraction
equals limit as n rightwards arrow infinity of 1 half open parentheses 1 third minus 1 fifth plus 1 fifth minus 1 over 7 plus 1 over 7 minus 1 over 9 plus........ plus fraction numerator 1 over denominator 2 n plus 1 end fraction minus fraction numerator 1 over denominator 2 n plus 3 end fraction close parentheses
equals limit as n rightwards arrow infinity of 1 half open parentheses 1 third minus fraction numerator 1 over denominator 2 n plus 3 end fraction close parentheses
equals 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator 2 n plus 3 minus 3 over denominator 3 left parenthesis 2 n plus 3 right parenthesis end fraction close parentheses
equals 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator 2 n over denominator 6 n plus 9 end fraction close parentheses
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 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator begin display style fraction numerator d blank over denominator d n end fraction end style 2 n over denominator begin display style fraction numerator d blank over denominator d n end fraction end style 6 n plus 9 end fraction close parentheses
equals 1 half cross times 2 over 6 equals 1 over 6

    L t subscript n not stretchy rightwards arrow straight infinity end subscript sum from n equals 1 to n of   open square brackets fraction numerator 1 over denominator left parenthesis 2 n plus 1 right parenthesis left parenthesis 2 n plus 3 right parenthesis end fraction close square brackets

    Maths-General
    L t subscript n not stretchy rightwards arrow straight infinity end subscript sum from n equals 1 to n of   open square brackets fraction numerator 1 over denominator left parenthesis 2 n plus 1 right parenthesis left parenthesis 2 n plus 3 right parenthesis end fraction close square brackets
equals limit as n rightwards arrow infinity of fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus....... plus fraction numerator 1 over denominator open parentheses 2 n plus 1 close parentheses left parenthesis 2 n plus 3 right parenthesis end fraction
equals limit as n rightwards arrow infinity of 1 half open parentheses 1 third minus 1 fifth plus 1 fifth minus 1 over 7 plus 1 over 7 minus 1 over 9 plus........ plus fraction numerator 1 over denominator 2 n plus 1 end fraction minus fraction numerator 1 over denominator 2 n plus 3 end fraction close parentheses
equals limit as n rightwards arrow infinity of 1 half open parentheses 1 third minus fraction numerator 1 over denominator 2 n plus 3 end fraction close parentheses
equals 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator 2 n plus 3 minus 3 over denominator 3 left parenthesis 2 n plus 3 right parenthesis end fraction close parentheses
equals 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator 2 n over denominator 6 n plus 9 end fraction close parentheses
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 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator begin display style fraction numerator d blank over denominator d n end fraction end style 2 n over denominator begin display style fraction numerator d blank over denominator d n end fraction end style 6 n plus 9 end fraction close parentheses
equals 1 half cross times 2 over 6 equals 1 over 6
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    Maths-

    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis comma plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction

    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis comma plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction
equals L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style 1 fourth end style n squared open parentheses n plus 1 close parentheses squared over denominator n squared open parentheses n squared plus 1 close parentheses end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n squared plus 2 n plus 1 over denominator n squared plus 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 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d n squared plus 2 n plus 1 over denominator d n end fraction end style over denominator begin display style fraction numerator d n squared plus 1 over denominator d n end fraction end style end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 n plus 2 over denominator 2 n end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n plus 1 over denominator n 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
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d n plus 1 over denominator d n end fraction end style over denominator begin display style fraction numerator d n over denominator d n end fraction end style end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript 1 over 1
equals 1 fourth

    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis comma plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction

    Maths-General
    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis comma plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction
equals L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style 1 fourth end style n squared open parentheses n plus 1 close parentheses squared over denominator n squared open parentheses n squared plus 1 close parentheses end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n squared plus 2 n plus 1 over denominator n squared plus 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 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d n squared plus 2 n plus 1 over denominator d n end fraction end style over denominator begin display style fraction numerator d n squared plus 1 over denominator d n end fraction end style end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 n plus 2 over denominator 2 n end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n plus 1 over denominator n 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
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d n plus 1 over denominator d n end fraction end style over denominator begin display style fraction numerator d n over denominator d n end fraction end style end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript 1 over 1
equals 1 fourth
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    An isosceles prism of angle 1200 has a refractive index of square root of 2 Two parallel monochromatic rays enter the prism parallel to each other in air as shown the rays emerging from the opposite face

    An isosceles prism of angle 1200 has a refractive index of square root of 2 Two parallel monochromatic rays enter the prism parallel to each other in air as shown the rays emerging from the opposite face

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    General
    physics-

    A light beam is travelling from Region I to Region IV (refer figure) The refractive index in Region I, II, III and IV are n subscript 0 end subscript comma fraction numerator n subscript 0 end subscript over denominator 2 end fraction comma fraction numerator n subscript 0 end subscript over denominator 6 end fraction text  and  end text fraction numerator n subscript 0 end subscript over denominator 8 end fraction, respectively The angle of incidence q for which the beam just misses entering Region IV is

    A light beam is travelling from Region I to Region IV (refer figure) The refractive index in Region I, II, III and IV are n subscript 0 end subscript comma fraction numerator n subscript 0 end subscript over denominator 2 end fraction comma fraction numerator n subscript 0 end subscript over denominator 6 end fraction text  and  end text fraction numerator n subscript 0 end subscript over denominator 8 end fraction, respectively The angle of incidence q for which the beam just misses entering Region IV is

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    General
    physics-

    A ray of light, travelling in a medium of refractive index 2, is incident onto the flat surface of a slab, the angle of incidence being 37° The refractive index m of the material of the slab varies with depth (x) from the top surface as mu open parentheses x close parentheses equals R open parentheses 2 minus x close parentheses, where x is in ‘m’ and R is 1 m–1 The maximum depth reached by the ray is text  (take  end text open sin invisible function application 37 to the power of ring operator end exponent almost equal to 0.6 close parentheses

    A ray of light, travelling in a medium of refractive index 2, is incident onto the flat surface of a slab, the angle of incidence being 37° The refractive index m of the material of the slab varies with depth (x) from the top surface as mu open parentheses x close parentheses equals R open parentheses 2 minus x close parentheses, where x is in ‘m’ and R is 1 m–1 The maximum depth reached by the ray is text  (take  end text open sin invisible function application 37 to the power of ring operator end exponent almost equal to 0.6 close parentheses

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