# Calculus: Limit at infinity

Suppose I have two continuous functions on $\mathbb R$, $f(x)$ and $g(x)$, such that $f(x)\leq g(x)$ for all $x\in \mathbb R$ and $\lim _{x \to\infty }g(x)=0$. Is the following true: $$\lim_{x \to\infty} f(x)\leq \lim_{x \to \infty} g(x) \>?$$

-

Yes, assuming the limit exists. If someone claims $\lim_{x \rightarrow\infty} f(x)=L\gt 0$, you can pick $M$ such that $g$ is within $L/2$ of 0 and $f$ is within $L/2$ of $L$. Then for $x \gt M, f(x) \gt g(x)$
(+1) I think we can get by without the limit of $f$ existing. Indeed, should it not be that, in fact, $\limsup_{x\to\infty} f(x) \leq \lim_{x\to\infty}g(x)$? And, then we can replace the left-hand side of the display equation in the question with a stronger version. – cardinal May 1 '11 at 0:13
Maybe I was being a little too subtle. Whether or not the limit of $f$ exists, under the given hypotheses, it is always true that $\limsup_{x\to\infty} f(x) \leq \lim_{x\to\infty} g(x)$ and this is a stronger statement. – cardinal May 1 '11 at 0:25