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Maths-

General

Easy

Question

$L t subscript left parenthesis x rightwards arrow 0 right parenthesis left parenthesis x 10 to the power of x minus x right parenthesis divided by left parenthesis 1 minus c o s invisible function application x right parenthesis equals$

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

hint Hint:

We are given a function. We have to find it's limit. We will denote the given function by f(x). While finding the limit we have to make sure that we don't get indeterminate form like 0/0 or infinity.

The correct answer is: $2 log invisible function application 10$

We have to find the following limit.
$limit as x rightwards arrow 0 of equals limit as x rightwards arrow 0 of fraction numerator x 10 to the power of x minus 1 over denominator 1 minus cos x end fraction space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator x left parenthesis 10 to the power of x minus 1 right parenthesis over denominator 1 end fraction cross times space fraction numerator 1 over denominator 1 space minus space cos x end fraction W e space h a v e space t w o space f o r m u l a s limit as x rightwards arrow 0 of space fraction numerator x squared over denominator 1 space minus space cos x end fraction space equals 2 limit as x rightwards arrow 0 of space fraction numerator a to the power of x minus 1 over denominator x end fraction equals log a space space space space space space left parenthesis a space i s space c o n s tan t right parenthesis S o comma space w e space w i l l space m u l t i p l y space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space b y space x squared$ $limit as x rightwards arrow 0 of f open parentheses x close parentheses equals limit as x rightwards arrow 0 of fraction numerator x open parentheses 10 to the power of x minus 1 close parentheses over denominator x squared end fraction cross times fraction numerator x squared over denominator 1 space minus space cos x end fraction space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator 10 to the power of x minus 1 over denominator x end fraction cross times fraction numerator x squared over denominator 1 space minus space cos x end fraction space space space space space space space space space space space space space space equals space log 10 space cross times space 2 space space space space space space space space space space space space space space space equals 2 log 10$
This is the required answer.

For such questions, we should know different formulae.

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