Exp运算法则

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$\begin{align} \lim_{h\to 0} \frac{\exp (x+h) - \exp (x)}{h} &=\exp (x) \lim_{h\to 0} \frac{\exp(h) - 1}{h}\\ &=\exp(x) \lim_{h\to 0}\frac{\sum_{n=0}^\infty \frac{h^n}{n!} -1}{h}\\ &= \exp(x) \lim_{h\to 0} \sum_{n=1}^\infty \frac{h^n}{n! \cdot h} \\ &= \exp(x)\sum_{n=0}^\infty \lim_{h\to 0} \frac{h^n}{(n + 1)!} \\ &= \exp(x) \cdot 1 = \exp(x) \end{align}$

$\begin{align} \lim_{h\to 0} \frac{\exp (x+h) - \exp (x)}{h} &=\exp (x) \lim_{h\to 0} \frac{\exp(h) - 1}{h}\\ &=\exp(x) \lim_{h\to 0}\frac{\sum_{n=0}^\infty \frac{h^n}{n!} -1}{h}\\ &= \exp(x) \lim_{h\to 0} \sum_{n=1}^\infty \frac{h^n}{n! \cdot h} \\ &= \exp(x)\sum_{n=0}^\infty \lim_{h\to 0} \frac{h^n}{(n + 1)!} \\ &= \exp(x) \cdot 1 = \exp(x) \end{align}$

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