# Solve $\lim_{x\rightarrow \infty} \left(x + \frac{\sin{x}}{x}\right)$

How to prove that $$\lim_{x\rightarrow \infty} \left(x + \frac{\sin{x}}{x}\right)$$ is equal to $$\infty$$?

I know that I couldn't use this: $$\lim_{x\rightarrow \infty} \left(x + \frac{\sin{x}}{x}\right)=\lim_{x\rightarrow \infty} x + \lim_{x\rightarrow \infty} \left(\frac{\sin{x}}{x} \right)= \infty + 0 =\infty .$$

• Here are the limit properties: tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx Apr 19 '20 at 9:20
• $\lvert\sin x\rvert\leqslant 1\implies\left\lvert\frac{\sin x}{x}\right\rvert\leqslant\frac{1}{\lvert x\rvert}\to 0$ as $x\to\pm\infty$ Apr 19 '20 at 9:48

## 2 Answers

Hint

For $$x>1$$:$$x+{\sin x\over x}>x-1$$

Here $$x\sim x+\frac{\sin x}{x}$$ for sufficiently large values of $$x.$$ So they are same at infinity.