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

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

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

If $f(x)$ is a differentiable function,then $\mathop {\lim }\limits_{x \to a} \frac{af(x) - xf(a)}{x - a}$ is

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