Definition: Differentiability (4.1)
A real function \(f\) is said to be differentiable at a point \(a \in \mathbb{R}\) if and only if \(f\) is defined on some open interval \(I\) containing \(a\) and $$ f'(a) := \lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h} $$ exists. In this case \(f'(a)\) is called the derivative of \(f\) at \(a\). To be further precises. For all \(\epsilon > \), there exists a \(\delta > 0\) such that if \(0 < |h| < \delta\), then $$ \left| \frac{f(a+h) - f(a)}{h} - L \right| < \epsilon $$ And \(L = f'(a)\) by definition.

Notice here that by a change of variable where we let \(h = x - a\), then we see that

$$ f'(a) := \lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h} = \lim\limits_{x \to a} \frac{f(x) - f(a)}{x - a} $$

References