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General

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Question

If $a greater than 0 text and end text Lt invisible function application left parenthesis x not stretchy rightwards arrow a right parenthesis open parentheses a to the power of x minus x to the power of 4 close parentheses divided by open parentheses x to the power of x minus a cubed close parentheses equals negative 1 text , then end text a equals$

0
1
-1
$\infty$

hint Hint:

In this question we will use L'Hôpital's rule, We will differentiate both numerator and denominator to find a.

The correct answer is: 1

In this question it is given that $limit as x minus greater than a of fraction numerator a to the power of x minus x to the power of a over denominator x to the power of x minus a to the power of a end fraction equals negative 1$
Step1: Applying L'Hôpital's rule
By differentiating both numerator and denominator.

=> $limit as x minus greater than a of fraction numerator a to the power of x ln open parentheses a close parentheses minus a x to the power of a minus 1 end exponent over denominator x to the power of x left parenthesis ln open parentheses x close parentheses plus 1 right parenthesis end fraction equals negative 1$

By putting the value of limit in the expression

$fraction numerator a to the power of a ln open parentheses a close parentheses minus a a to the power of a minus 1 end exponent over denominator a to the power of a left parenthesis ln open parentheses a close parentheses plus 1 right parenthesis end fraction equals negative 1$

=> $fraction numerator a to the power of a ln open parentheses a close parentheses minus a to the power of a over denominator a to the power of a left parenthesis ln open parentheses a close parentheses plus 1 right parenthesis end fraction equals negative 1$

=> $fraction numerator a to the power of a left parenthesis ln open parentheses a close parentheses minus 1 right parenthesis over denominator a to the power of a left parenthesis ln open parentheses a close parentheses plus 1 right parenthesis end fraction equals negative 1$

=> $left parenthesis ln open parentheses a close parentheses minus 1 right parenthesis equals negative 1 cross times left parenthesis ln open parentheses a close parentheses plus 1 right parenthesis$

=> $2 ln open parentheses a close parentheses equals 0$

=> $ln open parentheses a close parentheses equals 0$

=> $a equals 1$

So, the value $a$ is $1$ .

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