Maths-
General
Easy

Question

L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction

  1. 3 over 2
  2. 1 half
  3. 5 over 2
  4. 2 over 5

Hint:

In the question we will first simplify the equation using some trigonometric identities, then we will use L hospital rule,fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals fraction numerator f apostrophe left parenthesis x right parenthesis over denominator g apostrophe left parenthesis x right parenthesis end fraction, to get the answer.

The correct answer is: 3 over 2


    L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of fraction numerator tan cubed x minus begin display style fraction numerator sin squared x. space cos squared x over denominator cos squared x end fraction end style over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of fraction numerator tan cubed x left parenthesis 1 minus begin display style space cos squared x end style right parenthesis over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses
limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared 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
limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style left parenthesis 1 minus cos cubed x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x squared end fraction close parentheses
equals limit as x rightwards arrow 0 of fraction numerator 3 cos squared x. sin x over denominator 2 x end fraction
equals limit as x rightwards arrow 0 of 3 over 2 cos squared x. open parentheses fraction numerator sin x over denominator x end fraction close parentheses
equals 3 over 2 limit as x rightwards arrow 0 of cos squared x
equals 3 over 2 cos squared 0
equals 3 over 2
N o w comma limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses
equals 1 cross times 3 over 2 equals 3 over 2

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