Consider the following function: $$f(x) = x^2 \mid x \in \mathbb{R}, \ x \ne 0$$
The derivative at $x=0$ seems to want to be zero, in the same way that $\lim \limits_{x \to 0} f(x) = 0$
However, when I look at the definition of the derivative, this doesn't seem to work: $$f'(x) = \lim \limits_{\Delta x \to 0} \frac{f(x+\Delta x )-f(x)}{\Delta x}.$$ The function isn't defined at $f(x)$, so $f'(x)$ is also undefined. Would it make any sense to replace the $f(x)$ in the definition with $\lim \limits_{x \to 0} f(x) = 0$? Then, I suppose we'd have $\lim \limits_{x \to 0} f'(x) = 0$?
Would it be permissible? Would there be any point?