Suppose that $f$ is a real function and that $f'(a)$ exists:
$$ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}. $$
By replacing all instances of $h$ with $-h$, I can get an equivalent definition
$$ f'(a) = \lim_{-h \to 0} \frac{f(a-h) - f(a)}{-h}. $$
Question:
Is it right to say that I can replace $\lim_{-h \to 0}$ with $\lim_{h \to 0}$ in the second definition because $-h$ tending to $0$ is the same as saying $h$ tends to $0$ from the left/bottom?
Since $f'(a)$ exists, the left limit must be equal to the right limit and hence the switch is valid?