极限计算器

这个免费计算器会尝试求给定函数在给定点（包括无穷处）的极限（双侧或单侧，包含左极限和右极限），并显示解题步骤。

处理极限（包括未定式）可采用多种技巧：极限法则、重写与化简、洛必达法则、分母有理化、取自然对数等。

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Find the limit at:

If you need `oo`, type inf.

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Solution

Your input: find $$$\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x}$$$

Rewrite:

$${\color{red}{\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x}}} = {\color{red}{\lim_{x \to \infty}\left(1 + \frac{\sin{\left(x \right)}}{x}\right)}}$$

The limit of a sum/difference is the sum/difference of limits:

$${\color{red}{\lim_{x \to \infty}\left(1 + \frac{\sin{\left(x \right)}}{x}\right)}} = {\color{red}{\left(\lim_{x \to \infty} 1 + \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x}\right)}}$$

The limit of a constant is equal to the constant:

$$\lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} + {\color{red}{\lim_{x \to \infty} 1}} = \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} + {\color{red}{1}}$$

Since the absolute value of the sine is is not greater than $$$1$$$, then:

$$- \frac{1}{x} \leq \frac{\sin{\left(x \right)}}{x} \leq \frac{1}{x}$$

Taking the limits, we have that:

$$\lim_{x \to \infty}\left(- \frac{1}{x}\right) \leq \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} \leq \lim_{x \to \infty} \frac{1}{x}$$

$$0 \leq \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} \leq 0$$

Since the limits are equal, then, by the Squeeze Theorem:

$$\lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x}=0$$

Therefore,

$$\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x} = 1$$

Answer: $$$\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x}=1$$$

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逐步求极限

Solution