# 在无穷大方向上，函数可能存在水平渐近线和倾斜渐近线

## 二、解析

$$f(x)=a+g(x)$$

$$\lim \limits_{x \rightarrow+\infty} g(x)=0$$

$$\int_{0}^{+\infty} g(t) \mathrm{d} t=b$$

$$\lim \limits_{x \rightarrow+\infty} \int_{0}^{x} f(t) \mathrm{d} t=\lim \limits_{x \rightarrow+\infty}\left[\int_{0}^{x} a \mathrm{d} t+\int_{0}^{x} g(t) \mathrm{d} t\right] =$$

$$\lim \limits_{x \rightarrow+\infty} \int_{0}^{x} a \mathrm{d} t+b=+\infty$$

$$\lim \limits_{x \rightarrow+\infty} \frac{\int_{0}^{x} f(t) \mathrm{d} t}{x}=\lim \limits_{x \rightarrow+\infty} \frac{\int_{0}^{x} a \mathrm{d} t+\int_{0}^{x} g(t) \mathrm{d} t}{x}=$$

$$\lim \limits_{x \rightarrow 0} \frac{\int_{0}^{x} a \mathrm{d} t+b}{x}=\lim \limits_{x \rightarrow 0} \frac{a}{1}=a=k$$

$$\lim \limits_{x \rightarrow+\infty}[F(x)-a x]=$$

$$\lim \limits_{x \rightarrow+\infty}\left[\int_{0}^{x} f(t) \mathrm{d} t-a x\right]=$$

$$\lim \limits_{x \rightarrow+\infty} \int_{0}^{x}[f(t)-a] \mathrm{d} t=$$

$$\lim \limits_{x \rightarrow+\infty} \int_{0}^{x} g(t) \mathrm{d} t=$$

$$\int_{0}^{+\infty} g(t) \mathrm{d} t=b$$

$$y=a x+b$$