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Question

If $f left parenthesis a right parenthesis equals 2 comma f to the power of apostrophe left parenthesis a right parenthesis equals 1 comma g left parenthesis a right parenthesis equals negative 1 comma g to the power of factorial left parenthesis a right parenthesis equals 2 comma$ then the value of $L i m subscript left parenthesis x rightwards arrow a right parenthesis invisible function application left parenthesis g left parenthesis x right parenthesis f left parenthesis a right parenthesis minus g left parenthesis a right parenthesis f left parenthesis x right parenthesis right parenthesis divided by left parenthesis x minus a right parenthesis$ is
,

$- 5$
$1 / 5$
5
none of these

hint Hint:

In this question we are getting $0 over 0$ form. We will use L'Hôpital's rule to find the limit.

The correct answer is: 5

In this question we are given $f left parenthesis a right parenthesis equals 2$ , $f apostrophe left parenthesis a right parenthesis equals space 1$ and $g apostrophe left parenthesis a right parenthesis equals 2$ . We have to find $limit as x minus greater than a of fraction numerator g left parenthesis x right parenthesis f left parenthesis a right parenthesis minus g left parenthesis a right parenthesis f left parenthesis x right parenthesis over denominator x minus a end fraction$
Step1: Putting the value of the limit
By putting the value of the limit in the expression, we get, both numerator and denominator as zero. That is $0 over 0$ form.
Step2: Using L'Hôpital's rule
By differentiating both numerator and denominator we get,
$limit as x minus greater than a of fraction numerator g apostrophe left parenthesis x right parenthesis f left parenthesis a right parenthesis minus g left parenthesis a right parenthesis f apostrophe left parenthesis x right parenthesis over denominator 1 end fraction$

By putting the value of limits we get,
$limit as x minus greater than a of fraction numerator g apostrophe left parenthesis x right parenthesis f left parenthesis a right parenthesis minus g left parenthesis a right parenthesis f apostrophe left parenthesis x right parenthesis over denominator 1 end fraction equals 2 cross times 2 minus left parenthesis negative 1 right parenthesis cross times 1$
=> $5$
Hence, the value of the limit is $5$

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parallel

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parallel

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