## Calculus: Early Transcendentals (2nd Edition)

$$m_{tan}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a}m_{sec}$$ so we see that as we move $x$ towards $a$ the slope of the secant through $(x,f(x))$ and $(a,f(a))$ will approach the slope of the tangent at the point $(a,f(a))$.
The definition 1 says that the average rate of change of the function $f$ on the interval $[x,a]$ is $$m_{sec}=\frac{f(x)-f(a)}{x-a}$$ But this is also the slope of the secant line passing through $(x,f(x))$ and $(a,f(a))$. We also have that $$m_{tan}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a}m_{sec}$$ so we see that as we move $x$ towards $a$ the slope of the secant through $(x,f(x))$ and $(a,f(a))$ will approach the slope of the tangent at the point $(a,f(a))$.