Maths-
General
Easy

Question

L t left parenthesis h rightwards arrow 0 right parenthesis left parenthesis f left parenthesis 2 h plus 2 plus h squared right parenthesis minus f left parenthesis 2 right parenthesis right parenthesis divided by left parenthesis f left parenthesis h minus h squared plus 1 right parenthesis minus f left parenthesis 1 right parenthesis right parenthesis, given that f to the power of apostrophe left parenthesis 2 right parenthesis equals 6 and f to the power of apostrophe left parenthesis 1 right parenthesis equals 4

  1. is equal to 3
  2. does not exist
  3. is equal of 3 divided by 2
  4. is equal to left parenthesis negative 3 right parenthesis divided by 2

hintHint:

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

The correct answer is: is equal to 3


    In this question we are given f apostrophe left parenthesis 2 right parenthesis equals 6 and f apostrophe left parenthesis 1 right parenthesis equals 4 and we have to find limit as h minus greater than 0 of fraction numerator f left parenthesis 2 h plus 2 plus h squared right parenthesis minus f left parenthesis 2 right parenthesis over denominator f left parenthesis h minus h squared plus 1 right parenthesis minus f left parenthesis 1 right parenthesis end fraction
    Step1: Putting the value of limit
    By putting the value of limit we are getting 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 h minus greater than 0 of fraction numerator f apostrophe left parenthesis 2 h plus 2 plus h squared right parenthesis left parenthesis 2 plus 2 h right parenthesis over denominator f apostrophe left parenthesis h minus h squared plus 1 right parenthesis left parenthesis 1 minus 2 h right parenthesis end fraction

    By putting the value of limit we get
    limit as h minus greater than 0 of fraction numerator f apostrophe left parenthesis 2 right parenthesis left parenthesis 2 right parenthesis over denominator f apostrophe left parenthesis 1 right parenthesis left parenthesis 1 right parenthesis end fraction
    =>fraction numerator 2 cross times 6 over denominator 4 end fraction equals 3
    Hence, the value of the limit is 3.

     

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