引理

$$\lim\limits_{x \to \infty}(1 + \frac{1}{x}) ^ x = e\\ \lim\limits_{x \to 0}\sin{x} = x\\ $$

开证

$${f}(x) = \sin x$$ $${f}'(x) = \lim\limits_{\Delta{x} \to 0}\frac{{f}(x + \Delta{x}) - {f}(x)}{\Delta{x}}\\ = \lim\limits_{\Delta{x} \to 0}\frac{\sin(x + \Delta{x}) - \sin x}{\Delta{x}}\\ = \lim\limits_{\Delta{x} \to 0}\frac{\sin x \cos \Delta{x}+\cos x \sin \Delta{x} - \sin x}{\Delta{x}}\\ = \lim\limits_{\Delta{x} \to 0}\frac{\cos x \sin \Delta{x}}{\Delta{x}}\\ = \cos x\\ $$