三、洛必达法则

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1. 设 $x\to x_0$ （$x\to \infty$，或单侧极限）时，

(1)

$$\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}f(x)=0,\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}F(x)=0$$

(2) 在点 $x_0$ 的某个去心领域内（当$|x|>X$时），$f^{\prime }(x)$ 及 $F^{\prime }(x)$ 都存在且 $F^{\prime }(x)\ne 0$ ；

(3) $\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}\frac{f^{\prime }(x)}{F^{\prime }(x)}$ 存在（或为无穷大），则

$$\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}\frac{f(x)}{F(x)}=\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}\frac{f^{\prime }(x)}{F^{\prime }(x)}$$

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2. 设 $x\to x_0$ （$x\to \infty$，或单侧极限）时，

(1)

$$\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}f(x)=\infty ,\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}F(x)=\infty$$

(2) 在点 $x_0$ 的某个去心领域内（当$|x|>X$时），$f^{\prime }(x)$ 及 $F^{\prime }(x)$ 都存在且 $F^{\prime }(x)\ne 0$ ；

(3) $\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}\frac{f^{\prime }(x)}{F^{\prime }(x)}$ 存在（或为无穷大），则

$$\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}\frac{f(x)}{F(x)}=\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{lim}\frac{f^{\prime }(x)}{F^{\prime }(x)}$$