極限計算機
極限を段階的に計算
この無料の計算機は、手順を示しながら、指定された点（無限大を含む）における与えられた関数の極限（両側または片側、左極限および右極限を含む）を求めます。
極限（不定形を含む）を扱うために、さまざまな手法を用います：極限法則、式の書き換え・簡単化、ロピタルの定理、分母の有理化、自然対数を取る、など。
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: $$$\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x}=1$$$