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For function $f(x)$, let \begin{equation} f(c)=\lim_{x \to c} f(x). \end{equation} Under what conditions the following holds? \begin{equation} f'(c)=\lim_{x\to c} f'(x). \end{equation} For instance, let $f(x)=\frac{\sin(x)}{x}$, \begin{equation} f(0)=\lim_{x\to 0}\frac{\sin(x)}{x}=1. \end{equation} Is the following true? \begin{equation} f'(0)=\lim_{x\to 0}\left(\frac{\sin(x)}{x}\right)' \end{equation}

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    $\begingroup$ In your example, $f'(0)=0$. Is the limit of the derivative zero? In general though, derivatives may not be continuous. $\endgroup$ Nov 30, 2019 at 5:20
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    $\begingroup$ You have your equality if the limit on the right hand side exist (assuming that $f$ is differentiable everywhere). See here. $\endgroup$ Nov 30, 2019 at 5:22
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    $\begingroup$ It is true if $f$ is continuously differentiable at $x=c$. I'm not sure if that is an answer that is satisfactory for you though. $\endgroup$
    – Tucker
    Nov 30, 2019 at 6:43
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    $\begingroup$ For any function $g$, for $\lim_{x\to c}g(x)=g(c)$ to hold, it suffices that $g$ be continuous at $c$. So your question can be rephrased as "under what conditions is a derivative continuous?". A sufficient condition is the function twice differentiable, but this is not necessary. $\endgroup$
    – user65203
    Nov 30, 2019 at 8:27
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    $\begingroup$ Related: math.stackexchange.com/questions/257907/… $\endgroup$ Nov 30, 2019 at 9:40

1 Answer 1

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Recall that

$$f(c)=\lim_{x \to c} f(x)$$

is true if and only if $f(x)$ is continuous at $x=c$ and therefore

$$f'(c)=\lim_{x\to c} f'(x)$$

is true if and only if also $f'(x)$ is continuous at $x=c$.

For the example

$$f(x)=\frac{\sin(x)}{x} \implies f'(x)=\frac{x\cos x-\sin x}{x^2}$$

by $f(0)=1$ and $f'(0)=0$, are continuous at $x=0$ and both identities hold.

But the property is not true in general, e.g. $f(x)=|x|$.

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