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Question

(A) Given the limit: $\mathop {\lim }\limits_{x 	o a} \frac{xf(a) - af(x)}{x - a}$.To evaluate this,we add and subtract $af(a)$ in the numerator:$= \

Vedclass · Accepted Answer

$f(a) - a f'(a)$

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(A) Given the limit: $\mathop {\lim }\limits_{x 	o a} \frac{xf(a) - af(x)}{x - a}$.To evaluate this,we add and subtract $af(a)$ in the numerator:$= \mathop {\lim }\limits_{x 	o a} \frac{xf(a) - af(a) + af(a) - af(x)}{x - a}$$= \mathop {\lim }\limits_{x 	o a} \left[ \frac{f(a)(x - a)}{x - a} - \frac{a(f(x) - f(a))}{x - a} ight]$$= \mathop {\lim }\limits_{x 	o a} f(a) - a \mathop {\lim }\limits_{x 	o a} \frac{f(x) - f(a)}{x - a}$Since $f(x)$ is differentiable at $x = a$,$\mathop {\lim }\limits_{x 	o a} \frac{f(x) - f(a)}{x - a} = f'(a)$.Therefore,the limit is $f(a) - af'(a)$.

If $f(x)$ has a derivative at $x = a,$ then $\mathop {\lim }\limits_{x \to a} \frac{xf(a) - af(x)}{x - a}$ is equal to

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