单变量微积分学习笔记：重要极限（2）【1，13】

两个重要极限

\(\lim_{x \to 0}\frac{\sin(x)}{x} = 1\)
\(\lim_{x \to 0}(1+x)^{\frac{1}{x}} = e\)

证明

\(\lim_{x \to 0}\frac{\sin(x)}{x} = 1\)

\(x \to 0\)，弧趋于直线，即\(\lim_{x \to 0}\frac{\sin(x)}{x} = 1\)

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\(\lim_{x \to 0}(1+x)^{\frac{1}{x}} = e\)
证法1：导数定义
\(e^{\lim_{x \to 0}\frac{\ln(1+x)}{x}} = e^{\lim_{x \to 0}\frac{\ln(1+x)-\ln(1)}{x}} = e^{\lim_{\Delta x \to 0}\frac{\ln(1+\Delta x)-\ln(1)}{\Delta x}} = e^{\ln'(1)} = e\)

证法2：线性近似
\(x \to 0\)，\(e^{\frac{1}{x}ln(1+x)} \approx e^{\frac{1}{x}x} = e\)