Exercise 5.3.8

Question

Assume $f$ is continuous on an interval containing zero and differentiable for all $x 
eq 0$. If $\lim _{x ightarrow 0} f^{\prime}(x)=L$, show $f^{\prime}(0)$ exists and equals $L$.

Ulisse Mini · Accepted Answer

Using L'Hospital's rule: Let $g(x) = f(x) - f(0)$ (and note that they have the same derivatives and are both continuous), then
  $$f'(0) = g'(0) = \lim_{x \to 0} \frac{g(x)}{x} = \lim_{x \to 0}\frac{g'(x)}{1} = L$$
  (A modified function is necessary to ensure $\lim_{x \to 0} g(x) = 0$.)

Exercise 5.3.8

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Exercise 5.3.8

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