# Thread: limit of a function

1. ## limit of a function

hello,i tried to prove that the following limit in +and-infinity is equal to 0 but i couldn't,how can i prove that the limit of f(sin(x)) is a real number?thank you in advance for any help.(f is continuous in R)

2. ## Re: limit of a function

For all real $x$, $-1 \le \sin x \le 1$. By the extreme value theorem, a continuous function achieves a maximum and a minimum value over a closed interval, so there exists $c_1,c_2$ such that for all $x \in \mathbb{R}$, $f(\sin c_1) \le f(\sin x) \le f(\sin c_2)$.

Now you can use the Squeeze Theorem.

3. ## Re: limit of a function

It's worth pointing out that the extreme value theorem can be used only because both $\displaystyle f$ and $\displaystyle \sin x$ are continuous, and thus $\displaystyle f(\sin x)$ is also continuous.