Maths-
General
Easy

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
,

  1. negative 5
  2. 1 divided by 5
  3. 5
  4. none of these

hintHint:

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 2f 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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