What is the value for $\lim_{x\to\infty} \frac{\sin x}{x}$?

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  • Evaluating and proving $\lim_{x\to\infty}\frac{\sin x}x$

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Yes , the answer is $0$ .

One way to see this is by using the inequality :

$$\left |\frac{\sin x}{x}\right | \leq \frac{1}{x}$$ when $x>0$ (this happens because $|\sin x\ | \leq 1$ )

When $x \to \infty $ we have $\frac{1}{x} \to 0$ so the limit must be $0$ .

The range of $\sin(x)$ will always be a value between -1 and 1, no matter what the input. However, there is no such restriction on the denominator. Therefore, if your numerator is restricted to a finite value, and your denominator is not, as the denominator goes to infinity the value of the whole expression will go to zero.