# 数字零和极限零有什么区别？

## 二、正文

\textcolor{springgreen}{ \begin{aligned} & \lim_{x \rightarrow 0} \frac{2-2}{x} \\ \\ = & \lim_{x \rightarrow 0} \frac{0}{x} \\ \\ = & \lim_{x \rightarrow 0} \frac{0}{x ^{2}} \\ \\ = & \lim_{x \rightarrow 0} \frac{0}{x ^{3}} \\ \\ = & 0 \end{aligned} }

\textcolor{springgreen}{ \begin{aligned} \lim_{x \rightarrow 0} \frac{x ^{2}}{x} = & \lim_{x \rightarrow 0} x \rightarrow 0 \\ \\ \lim_{x \rightarrow 0} \frac{x}{x ^{2}} = & \lim_{x \rightarrow 0} \frac{1}{x} \rightarrow \infty \\ \\ \lim_{x \rightarrow 0} \frac{2x}{x} = & 2 \end{aligned} }

\textcolor{springgreen}{ \begin{aligned} K ^{0} = & 1 \\ \\ \lim_{x \rightarrow 0} K ^{x} = & 1 \end{aligned} }

$$\textcolor{springgreen}{ \lim_{x \rightarrow 0} x ^{x} = 1 } \tag{1}$$

\textcolor{springgreen}{ \begin{aligned} & \lim_{x \rightarrow 0} \left( 1 + \frac{1}{x} \right) ^{x} = \mathrm{e} \\ \\ & \lim_{x \rightarrow \infty} \left( 1 + x \right) ^{\frac{1}{x}} = \mathrm{e} \end{aligned} }

$$\textcolor{springgreen}{ 0! = 1! = 1 }$$

$$\lim_{n \rightarrow 0} n! = 0$$

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