Question

If f is differentiable at r = a, then prove that
$Lt with straight x rightwards arrow straight a below fraction numerator straight x space straight f left parenthesis straight a right parenthesis minus straight a space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals straight f left parenthesis straight a right parenthesis equals straight a space straight f left parenthesis straight a right parenthesis$

Solution

straight L. straight H. straight S equals Lt with straight x rightwards arrow straight a below fraction numerator straight x space straight f left parenthesis straight a right parenthesis minus straight a space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals Lt with straight x rightwards arrow straight a below fraction numerator straight x space straight f left parenthesis straight a right parenthesis minus straight a space straight f left parenthesis straight a right parenthesis plus straight a space straight f left parenthesis straight a right parenthesis minus straight a space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction space space space space space space space space space equals Lt with straight x rightwards arrow straight a below fraction numerator left parenthesis straight x minus straight a right parenthesis space straight f left parenthesis straight a right parenthesis minus straight a left curly bracket straight f left parenthesis straight x right parenthesis minus straight f left parenthesis straight a right parenthesis right curly bracket over denominator straight x minus straight a end fraction equals Lt with straight x rightwards arrow straight a below open square brackets fraction numerator left parenthesis straight x minus straight a right parenthesis space straight f left parenthesis straight a right parenthesis over denominator straight x minus straight a end fraction minus straight a fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight x minus straight a end fraction close square brackets space space space space space space space space space equals Lt with straight x rightwards arrow straight a below straight f left parenthesis straight a right parenthesis minus straight a space Lt with straight x rightwards arrow straight a below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight x minus straight a end fraction equals straight f left parenthesis straight a right parenthesis minus straight a space straight f left parenthesis straight a right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because straight f left parenthesis straight x right parenthesis space is space differentiable space aty space straight x equals straight a right square bracket

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