The Tangent Line

If $f$ is defined on an open interval containing $c$ , and if the limit

$$\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x\to 0}\frac{f(c+\Delta x)-f(c)}{\Delta x}=m$$

exists, then the line passing through $(c , f(c))$ with slope $m$ is the tangent line to the graph of $f$ at the point $(c , f(c)).$