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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: $$\lim_{x\rightarrow \infty} \left(x + \frac{\sin{x}}{x}\right)=\lim_{x\rightarrow \infty} x + \lim_{x\rightarrow \infty} \left(\frac{\sin{x}}{x} \right)= \infty + 0 =\infty .$$

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Hint

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

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

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