[tex]\begin{aligned}f'(x) &= \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} \\&= \lim_{h \to 0} \dfrac{(x+h) - x}{h} \\&=\lim_{h \to 0} \dfrac{h}{h} \\&=\lim_{h \to 0} (1) && (\text{\footnotesize since $h/h = 1$ for $h\ne 0$})\\&= 1\end{aligned}[/tex]
h/h = 1 is valid for h ≠ 0. The limit does not care about h at 0; it only cares about the values around it.
