Derivative of a general linear function via the limit definition

First compute $f(x+h) = m(x+h) + b = mx + mh + b$. Then form the difference $f(x+h) - f(x) = (mx + mh + b) - (mx + b) = mh$. Dividing by $h$ gives the difference quotient $\dfrac{f(x+h)-f(x)}{h} = \dfrac{mh}{h} = m$, valid for $h \neq 0$.

Since the difference quotient simplifies to the constant $m$ for all $h \neq 0$, taking the limit as $h \to 0$ gives $f'(x) = \lim_{h \to 0} m = m$. This confirms that a linear function has constant slope $m$ everywhere, matching the geometric fact that its tangent line coincides with the line itself.

The h canceled because the difference quotient simplified to just m, meaning the average rate of change was already constant before any limit was taken. This happens because f is linear, so its slope never changes between any two points. As a result, the limit as h approaches zero equals m trivially, since the expression no longer depends on h at all.