Limits and Derivatives

Limits and Derivatives

Question

If f is derivable at x=a, evaluate space space space space space space space space space stack space lim with straight x rightwards arrow straight a below fraction numerator xf left parenthesis straight a right parenthesis minus af left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction.

Answer

Since  f is derivable at x =a

∴       space space space space space space space space space space space space space space straight f apostrophe left parenthesis straight a right parenthesis space space space space equals space limit as straight h rightwards arrow straight a of fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction
Now,    space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator xf left parenthesis straight a right parenthesis minus af left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals limit as straight h rightwards arrow 0 of fraction numerator left parenthesis straight a plus straight h right parenthesis space straight f left parenthesis straight a right parenthesis space minus af left parenthesis straight a plus straight h right parenthesis over denominator straight a plus straight h minus straight a end fraction

                                                                                 [By putting x =a +h as xrightwards arrowa, h rightwards arrow0]


space space space space space space space space space space space space space space space space space equals space space space limit as straight h rightwards arrow 0 of fraction numerator af left parenthesis straight a right parenthesis plus hf left parenthesis straight a right parenthesis minus af left parenthesis straight a plus straight h right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator straight a left square bracket straight f left parenthesis straight a right parenthesis minus straight f left parenthesis straight a plus straight h right parenthesis plus right square bracket plus hf left parenthesis straight a right parenthesis over denominator straight h end fraction

    space space space space space space space space space space space space space equals negative straight a space limit as straight h rightwards arrow 0 of open square brackets fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction plus straight f left parenthesis straight a right parenthesis close square brackets
     space space space space space space space space space space space space space space equals space minus straight a space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction plus limit as straight h rightwards arrow 0 of straight f left parenthesis straight a right parenthesis

                 = -af'(a) + f(a)                                                                           [By using (1)]
                  =  f(a)-af'(a)                                                             

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