Let f(x) be differentiable at x = h. Then lim | vedclass

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(A) Let $L = \lim_{x 	o h} \frac{(x + h)f(x) - 2hf(h)}{x - h}$.Since the limit is in the indeterminate form $\frac{0}{0}$ as $x 	o h$,we apply $L'

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$f(h) + 2hf'(h)$

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(A) Let $L = \lim_{x 	o h} \frac{(x + h)f(x) - 2hf(h)}{x - h}$.Since the limit is in the indeterminate form $\frac{0}{0}$ as $x 	o h$,we apply $L'	ext{Hospital's rule}$ by differentiating the numerator and the denominator with respect to $x$.Numerator derivative: $\frac{d}{dx} [(x + h)f(x) - 2hf(h)] = (x + h)f'(x) + f(x)(1) - 0 = (x + h)f'(x) + f(x)$.Denominator derivative: $\frac{d}{dx} (x - h) = 1$.Now,evaluate the limit as $x 	o h$:$L = \lim_{x 	o h} [(x + h)f'(x) + f(x)] = (h + h)f'(h) + f(h) = 2hf'(h) + f(h)$.Thus,the correct option is $A$.

Let $f(x)$ be differentiable at $x = h$. Then $\lim_{x \to h} \frac{(x + h)f(x) - 2hf(h)}{x - h}$ is equal to

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