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Maths-

General

Easy

Question

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x 10 to the power of x minus x over denominator 1 minus cos space x end fraction equals$

$\log 10$
$3 \log 10$
$2 \log 10$
$4 \log 10$

hint Hint:

In the given question if we put x=0 in the function the value is indeterminate so first we will simplify the given function, separate the trigonometric and logarithmic function then use some trigonometric identities and then use the theorems to find the value of the limit.

The correct answer is: $2 log space 10$

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x 10 to the power of x minus x over denominator 1 minus cos space x end fraction equals$ $limit as x rightwards arrow 0 of fraction numerator x.10 to the power of x minus x over denominator 1 minus cos x end fraction$
$equals limit as x rightwards arrow 0 of fraction numerator x to the power of 2 space end exponent open parentheses 10 to the power of x minus 1 close parentheses over denominator left parenthesis 1 minus cos space x right parenthesis space x end fraction equals limit as x rightwards arrow 0 of fraction numerator x squared over denominator 1 minus cos space x end fraction space limit as x rightwards arrow 0 of fraction numerator 10 to the power of x minus 1 over denominator x end fraction space space space space space space open square brackets limit as x rightwards arrow 0 of fraction numerator a to the power of x minus 1 over denominator x end fraction equals log space a close square brackets equals log space 10 space limit as x rightwards arrow 0 of fraction numerator x squared over denominator 2 sin squared begin display style x over 2 end style end fraction space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets space equals log space 10 space 1 half limit as x rightwards arrow 0 of fraction numerator begin display style x over 2 end style over denominator sin space begin display style x over 2 end style end fraction fraction numerator begin display style x over 2 end style.4 over denominator sin begin display style x over 2 end style end fraction equals log space 10 fraction numerator space 1 over denominator 2 end fraction space 4 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets limit as x rightwards arrow 0 of fraction numerator x over denominator sin space x end fraction equals 1 close square brackets equals 2 log space 10$

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