calculus - Two simple statements about continuous and monotonic functions

Suppose $f: \mathbb{R} \to \mathbb{R}$. I must determine whether the following statements are true:

1. If $f$ is continuous on $\mathbb{R}$ and not bounded then $\lim_{x\to \infty} f(x)$ is either $\infty$ or $-\infty$.




2. If $f$ is strictly increasing and is not bounded from below then $\lim_{x\to -\infty} f(x)=-\infty$

I believe that both are true. In 1. I think that if the function is not bounded and has $\lim_{x\to \infty} f(x)=L$ then it must tend to $\pm$infinity at some point. But if $f$ tends to $\pm \infty$ at any point then it can't be continuous there. So it must tend to $\pm \infty$ at $\infty$.

In 2. I can't find a counterexample but I have a hard time proving the statement. Intuitively, I think that it is true. Am I correct?

Answer

1. Consider $f(x)=x\sin x$.


2. This is true. Hint for a proof: For any $L\in \mathbb R$, there exists $c\in \mathbb R$ such that $f(c) < L$, since $f$ is not bounded below. Also, if $f(c)< L$, then $f(x) < L$ for all $x\leq c$, since $f$ is strictly increasing.