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Maths-

General

Easy

Question

$L t subscript left parenthesis x rightwards arrow 0 right parenthesis invisible function application left parenthesis 6 to the power of x minus 3 to the power of x minus 2 to the power of x plus 1 right parenthesis divided by x squared$

$\log 3 \times \log 6$
$\log 2 \times \log 3$
$\log 2 \times \log 6$
$\log 3 + \log 2$

hint Hint:

We are given a function. We have to find the limit of the function.

The correct answer is: $log invisible function application 2 cross times log invisible function application 3$

The given function is $f open parentheses x close parentheses equals fraction numerator 6 to the power of x minus 3 to the power of x minus 2 to the power of x plus 1 over denominator x squared end fraction$
We have to take limit of the function.
Dbs $limit as x rightwards arrow 0 of f open parentheses x close parentheses equals limit as x rightwards arrow 0 of fraction numerator 6 to the power of x minus 3 to the power of x minus 2 to the power of x plus 1 over denominator x squared end fraction I f space w e space s e e space w e space g e t space z e r o space f o r space n u m e r a t o r space a n d space d e n o m i n a t o r i f space w e space s u b s t i t u t e space x space equals space 0 S o comma space w e space w i l l space r e a r r a n g e space t h e space e q u a t i o n w e space w i l l space b e space u sin g space t h e space f o r m u l a limit as x rightwards arrow 0 of fraction numerator a to the power of x minus 1 over denominator x end fraction space equals space log a$
$limit as x rightwards arrow 0 of f open parentheses x close parentheses equals limit as x rightwards arrow 0 of fraction numerator 6 to the power of x minus 3 to the power of x minus 2 to the power of x plus 1 over denominator x squared end fraction space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator 2 to the power of x 3 to the power of x minus 3 to the power of x minus 2 to the power of x plus 1 over denominator x squared end fraction space space space space... left curly bracket 6 to the power of x space equals left parenthesis 2.3 right parenthesis to the power of x space equals 2 to the power of x end exponent 3 to the power of x right curly bracket space space space space space space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator 3 to the power of x left parenthesis 2 to the power of x minus 1 right parenthesis minus 1 left parenthesis 2 to the power of x minus 1 right parenthesis over denominator x squared end fraction space space space space space space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator left parenthesis 3 to the power of x minus 1 right parenthesis left parenthesis 2 to the power of x minus 1 right parenthesis over denominator x squared end fraction space space space space space space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator 3 to the power of x minus 1 over denominator x end fraction cross times fraction numerator 2 to the power of x minus 1 over denominator x end fraction space space space space space space space space space space space equals log 3 space cross times space log 2$ This is the required answer.

For such questions, we should be know different formulas of limit.

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