TITLE: Question about the derivative definition
QUESTION [13 upvotes]: The derivative at a point $x$ is defined as:
$\lim\limits_{h\to0} \frac{f(x+h) - f(x)}h$
But if $h\to0$, wouldn't that mean:
$\frac{f(x+0) - f(x)}0 = \frac0{0}$ which is undefined?

REPLY [2 votes]: It might be helpful to consider the alternate notation in this case as written in: https://en.wikipedia.org/wiki/Derivative
Consider $\lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}$ where $y = f(x)$. Under this notation you'll note that the denominator is not a variable per say, but a difference. In your notation $h = \Delta x$. This notation can be misleading, leading you to conclude that $h$ is a variable that can be substituted. In fact (as eluded to by others) $\Delta x = 0$ can't be "reached", hence the need for a limit.