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Question

If $G left parenthesis x right parenthesis equals negative √ left parenthesis 25 minus x squared right parenthesis$ then $L i m subscript left parenthesis x rightwards arrow 1 right parenthesis invisible function application left parenthesis G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis right parenthesis divided by left parenthesis x minus 1 right parenthesis$ has the value

$1 / 24$
$1 / 5$
$- \sqrt 24$
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: none of these

In this question we are given $G left parenthesis x right parenthesis space equals space minus square root of 25 minus x squared end root$ and we have to find the limit $limit as x minus greater than 1 of fraction numerator G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis over denominator x minus 1 end fraction$
Step1: Putting the value of limit
By putting the value of 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,

=> $limit as x minus greater than 1 of fraction numerator G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis over denominator x minus 1 end fraction$

$limit as x minus greater than 1 of G apostrophe left parenthesis x right parenthesis$

By putting the value of limit we get,

$G left parenthesis x right parenthesis space equals space minus square root of 25 minus x squared end root$

$G apostrophe left parenthesis x right parenthesis space equals space fraction numerator negative 2 x over denominator 2 square root of 25 minus x squared end root end fraction$

$limit as x minus greater than 1 of G apostrophe left parenthesis x right parenthesis space equals space limit as x minus greater than 1 of fraction numerator negative 2 x over denominator 2 square root of 25 minus x squared end root end fraction$

$G apostrophe left parenthesis 1 right parenthesis equals space fraction numerator 1 over denominator square root of 24 end fraction$

Related Questions to study

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parallel

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