Maths-
General
Easy

Question

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of 1 plus x end root minus square root of 1 plus x squared end root over denominator square root of 1 minus x squared end root minus square root of 1 minus x end root end fraction

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

The correct answer is: 1

Related Questions to study

General
Maths-

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction
using L hospital theorem,
equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d square root of 1 plus x plus x squared end root over denominator d x end fraction end style minus begin display style fraction numerator d blank over denominator d x end fraction end style 1 over denominator begin display style fraction numerator d x over denominator d x end fraction end style end fraction
equals limit as x rightwards arrow 0 of fraction numerator 1 plus 2 x over denominator 2 square root of 1 plus x plus x squared end root end fraction
equals fraction numerator 1 plus 2 cross times 0 over denominator 2 square root of 1 plus 0 plus 0 end root end fraction
equals 1 half

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction

Maths-General
L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction
using L hospital theorem,
equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d square root of 1 plus x plus x squared end root over denominator d x end fraction end style minus begin display style fraction numerator d blank over denominator d x end fraction end style 1 over denominator begin display style fraction numerator d x over denominator d x end fraction end style end fraction
equals limit as x rightwards arrow 0 of fraction numerator 1 plus 2 x over denominator 2 square root of 1 plus x plus x squared end root end fraction
equals fraction numerator 1 plus 2 cross times 0 over denominator 2 square root of 1 plus 0 plus 0 end root end fraction
equals 1 half
General
Maths-

Lt subscript x not stretchy rightwards arrow 3 end subscript fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction

Lt subscript x not stretchy rightwards arrow 3 end subscript fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction
using L hospital theorem,
limit as x rightwards arrow 3 of fraction numerator begin display style fraction numerator d x cubed minus 8 x squared plus 45 over denominator d x end fraction end style over denominator begin display style fraction numerator d 2 x squared minus 3 x minus 9 over denominator d x end fraction end style end fraction
equals limit as x rightwards arrow 3 of fraction numerator 3 x squared minus 16 x over denominator 4 x minus 3 end fraction
equals fraction numerator 3 cross times 9 minus 16 cross times 3 over denominator 4 cross times 3 minus 3 end fraction
equals fraction numerator 27 minus 48 over denominator 12 minus 3 end fraction
equals fraction numerator negative 21 over denominator 9 end fraction
equals fraction numerator negative 7 over denominator 3 end fraction

Lt subscript x not stretchy rightwards arrow 3 end subscript fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction

Maths-General
Lt subscript x not stretchy rightwards arrow 3 end subscript fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction
using L hospital theorem,
limit as x rightwards arrow 3 of fraction numerator begin display style fraction numerator d x cubed minus 8 x squared plus 45 over denominator d x end fraction end style over denominator begin display style fraction numerator d 2 x squared minus 3 x minus 9 over denominator d x end fraction end style end fraction
equals limit as x rightwards arrow 3 of fraction numerator 3 x squared minus 16 x over denominator 4 x minus 3 end fraction
equals fraction numerator 3 cross times 9 minus 16 cross times 3 over denominator 4 cross times 3 minus 3 end fraction
equals fraction numerator 27 minus 48 over denominator 12 minus 3 end fraction
equals fraction numerator negative 21 over denominator 9 end fraction
equals fraction numerator negative 7 over denominator 3 end fraction
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Oxygen atom of ether is:

Oxygen atom of ether is:

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A long current carrying wire, carrying current such that it is flowing out from the plane of paper, is placed at O A steady state current is flowing in the loop ABCD

   

A long current carrying wire, carrying current such that it is flowing out from the plane of paper, is placed at O A steady state current is flowing in the loop ABCD

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A loop carrying current I lies in the x-y plane as shown in fig The unit vector k^ is coming out of the plane of the paper The magnetic moment of the current loop is

   

A loop carrying current I lies in the x-y plane as shown in fig The unit vector k^ is coming out of the plane of the paper The magnetic moment of the current loop is

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A particle originally at rest at the highest point of a smooth vertical circle is slightly displaced. It will leave the circle at a vertical distance h below the highest point such that

   

A particle originally at rest at the highest point of a smooth vertical circle is slightly displaced. It will leave the circle at a vertical distance h below the highest point such that

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In figure, a particle is placed at the highest point A of a smooth sphere of radius r. It is given slight push, and it leaves the sphere at B, at a depth h vertically below A such that h is equal to

If v velocity acquired at B, then v2=2gh The particle will leave the sphere at B, when mv2rmgcosθ 2ghr=g(r-h)r, Which gives h=r3    

In figure, a particle is placed at the highest point A of a smooth sphere of radius r. It is given slight push, and it leaves the sphere at B, at a depth h vertically below A such that h is equal to

physics-General
If v velocity acquired at B, then v2=2gh The particle will leave the sphere at B, when mv2rmgcosθ 2ghr=g(r-h)r, Which gives h=r3    
General
Maths-

Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of tan space x end exponent minus e to the power of x over denominator tan space x minus x end fraction

space space space space Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of tan space x end exponent minus e to the power of x over denominator tan space x minus x end fraction
equals space Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of x left parenthesis e to the power of tan space x minus x end exponent minus 1 right parenthesis over denominator tan space x minus x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript space e to the power of x
equals e to the power of 0
equals 1.

Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of tan space x end exponent minus e to the power of x over denominator tan space x minus x end fraction

Maths-General
space space space space Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of tan space x end exponent minus e to the power of x over denominator tan space x minus x end fraction
equals space Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of x left parenthesis e to the power of tan space x minus x end exponent minus 1 right parenthesis over denominator tan space x minus x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript space e to the power of x
equals e to the power of 0
equals 1.
General
Maths-

Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x times 2 to the power of x minus x over denominator 1 minus cos space x end fraction

space space space Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x times 2 to the power of x minus x over denominator 1 minus cos space x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x times left parenthesis 2 to the power of x minus 1 right parenthesis over denominator 1 minus cos space x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x squared over denominator 1 minus cos space x end fraction. fraction numerator 2 to the power of x minus 1 over denominator x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript space 4 cross times fraction numerator begin display style x squared over 4 end style over denominator 2 sin squared begin display style x over 2 end style end fraction. fraction numerator 2 to the power of x minus 1 over denominator x end fraction
equals 2 space log space 2.

Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x times 2 to the power of x minus x over denominator 1 minus cos space x end fraction

Maths-General
space space space Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x times 2 to the power of x minus x over denominator 1 minus cos space x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x times left parenthesis 2 to the power of x minus 1 right parenthesis over denominator 1 minus cos space x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x squared over denominator 1 minus cos space x end fraction. fraction numerator 2 to the power of x minus 1 over denominator x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript space 4 cross times fraction numerator begin display style x squared over 4 end style over denominator 2 sin squared begin display style x over 2 end style end fraction. fraction numerator 2 to the power of x minus 1 over denominator x end fraction
equals 2 space log space 2.
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A small sphere is hung by a string fixed to a wall. The sphere is pushed away from the wall by a stick. The force acting on the sphere are shown in figure. Which of the following statements is wrong?

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A small sphere is hung by a string fixed to a wall. The sphere is pushed away from the wall by a stick. The force acting on the sphere are shown in figure. Which of the following statements is wrong?

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parallel

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