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Maths-

General

Easy

Question

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

$\frac{3}{2}$
$\frac{1}{2}$
$\frac{5}{2}$
$\frac{2}{5}$

hint 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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