Why does h approach 0 and not f(x + h) - f(x) when taking the derivative?

Question

Why does the blue section approach 0 and not the black one? Also, if h would approach Zero, wouldn't f(x+h)-f(x) too?

Answer 1

The comments have basically answered this, but to make it explicit:

When taking the derivative, we're interested in the limit of the ratio $\frac{f(x+h)-f(x)}{h}$ as $h$ approaches $0$.
If $f$ is continuous at $x$, then both the numerator and the denominator will approach $0$ as $h$ approaches $0$, but the ratio frequently will not.

Explicitly having $f(x+h)-f(x)$ approach $0$ probably wouldn't be useful.
We want the ratio; if we choose a value of $h$ then it's "clear" how to get both the numerator and denominator. If we choose a value of the numerator we might not have a way for calculating the denominator $h$, and in fact a value of $h$ satisfying our choice isn't guaranteed to exist.