1. ## Max and Mins

Suppose that $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous on $\mathbb{R}$ and that $\lim_{x\to\infty}f = 0$.
How would you prove that f is bounded on R and attains either a max or a min on R?

2. Originally Posted by frenchguy87
Suppose that $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous on $\mathbb{R}$ and that $\lim_{x\to\infty}f = 0$.
How would you prove that f is bounded on R and attains either a max or a min on R?
It doesn't have to. $f(x)=e^{-x}$

3. Originally Posted by Drexel28
It doesn't have to. $f(x)=e^{-x}$

Perhaps he meant the limit as x approaches both + and - sends f(x) to 0?

4. Originally Posted by southprkfan1
Perhaps he meant the limit as x approaches both + and - sends f(x) to 0?
If so, I will help with the bounded part. The other is up to you. Since $\lim_{x\to\infty}f(x)=\lim_{x\to-\infty}f(x)=0$ there exists some $T>0$ such that $x\in(-\infty,-T)\cup(T,\infty)$ implies that $|f(x)|<1$. But, then we have that $f:[-T,T]\mapsto\mathbb{R}$ is bounded (since $[-T,T]$ is compact and $f$ continuous). Thus, $\left|f(x)\right|\leqslant M,\text{ }x\in[-T,T]$ for some $M\in\mathbb{R}$. It readily follows that $|f(x)|\leqslant\max\left\{1,M\right\}$ for all $x\in\mathbb{R}$