Extreme Value Theorem on an Unbounded Domain

Question

Given a smooth function $f(x)\colon \mathbb{R} \to \mathbb{R} $. Suppose $$\lim_{x\to -\infty} f(x) = 0, $$ and $$ \lim_{x\to \infty} f(x) = 0. $$ Can we claim that $f$ is bounded?

Answer 1

Hint: Turn this "picture" into a proof

If $x$ is sufficiently large, say $x>M$ then $f(x)$ is small.

If $x$ is sufficiently small, say $x<-N$ then $f(x)$ is small.

The interval $[-N,M]$ is closed, $f$ is continuous and therefore $f$ is bounded on this interval.

So it's either small, or bounded, or small. Therefore it is bounded.

Now make this formal