can somebody tell me the proof to $\displaystyle \frac{\delta y}{\delta x}$ represents the gradient of tangent?

The conclusion is $\displaystyle \frac{dy}{dx}$= $\displaystyle \lim(\delta x\rightarrow 0)(\frac{\delta y}{\delta x})$.

can explain why must it be $\displaystyle \delta x\rightarrow 0$?