TITLE: L'Hôpital's rule and Difference Quotients
QUESTION [5 upvotes]: Consider the general difference quotient for a function $f(x)$ that is differentiable at $x = a$:
$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$
Since both the numerator and denominator of the difference quotient are differentiable and approach 0, it seems like we should be able to apply l'Hospital's rule, differentiating with respect to $x$:
\begin{align}
f'(a) & = \lim_{x \to a} \, \frac{f(x) - f(a)}{x - a} \\
& = \lim_{x \to a} \, f'(x), \text{ given we fix $a$ and have assumed $f$ is differentiable}
\end{align}
But this seems to imply that $f'(x)$ is continuous or $f(x)$ is continuously differentiable at $x = a$. 
I don't understand where this additional condition comes from: I suspect the problem may be in applying l'Hospital's rule (perhaps it's circular?). But to use l'Hospital's and know the limit existed, it was sufficient to assume $f(x)$ was differentiable at $x = a$. Nowhere did we have to make the assumption that the function be continuously differentiable.
I must be overlooking some small detail, but I just cannot put my finger on it. Any help would be appreciated.

REPLY [3 votes]: This is a vey nice case that shows us how important are assumptions. In De l'Hospital's theorem, we must assume that both the numerator and the denominator are differentiable in a neighborhood of $a$: this is much more than assuming the differentiability at $a$.
Bu what is even more important, is that this theorem states that IF
$$
\lim_{x\to a} f'(x) \quad\text{exists}
$$
THEN also
$$
\lim_{x \to a} \frac{f(x)-f(a)}{x-a} \quad\text{exists and has the same value}.
$$
In other words, De l'Hospital's theorem applies only if $f'$ is a continuous function at $a$. We should not be surprised, since De l'Hospital's theorem cannot be as strong as the very definition of derivative.