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Continuity And Differentiability

Question
CBSEENMA12034655

Let f be differentiable at a, then evaluate

Lt with straight x rightwards arrow straight a below fraction numerator straight x squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction

Solution
Put space straight x equals straight a plus straight h space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow straight a
therefore space Lt with straight x rightwards arrow straight a below fraction numerator straight x squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator left parenthesis straight a plus straight h right parenthesis squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight a plus straight h right parenthesis over denominator straight a plus straight h minus straight a end fraction
space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator left parenthesis straight a squared plus 2 space straight a space straight h plus straight h right parenthesis squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight a plus straight h right parenthesis over denominator straight h end fraction
space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator straight a squared space straight f left parenthesis straight a right parenthesis plus 2 space straight a space straight h space straight f left parenthesis straight a right parenthesis plus straight h squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight a plus straight h right parenthesis over denominator straight h end fraction

space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator straight h left curly bracket 2 space straight a space straight f left parenthesis straight a right parenthesis plus straight h space straight f left parenthesis straight a right parenthesis right curly bracket minus straight a squared left curly bracket straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis right curly bracket over denominator straight h end fraction
space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below open square brackets 2 space straight a space straight f left parenthesis straight a right parenthesis plus straight h space straight f left parenthesis straight a right parenthesis minus straight a squared 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 close square brackets
space space space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow straight a below 2 space straight a space straight f left parenthesis straight a right parenthesis plus Lt with straight h rightwards arrow 0 below straight h space straight f left parenthesis straight a right parenthesis minus straight a squared space Lt with straight h rightwards arrow 0 below 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
space space space space space space space space space space space space space space space space space equals 2 space straight a space straight f left parenthesis straight a right parenthesis plus 0 straight f left parenthesis straight a right parenthesis minus straight a squared straight f left parenthesis straight a right parenthesis equals 2 space straight a space straight f left parenthesis straight a right parenthesis minus straight a squared straight f left parenthesis straight a right parenthesis.

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