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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: $$ \lim_{x \to\infty} f(x)\leq \lim_{x \to \infty} g(x) \>?$$

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1 Answer 1

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 $x \gt M, f(x) \gt g(x)$

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(+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
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@cardinal: but if the limit doesn't exist, it doesn't make sense to ask if it is less than something else. You are right about lim sup f by the same style of argument. –  Ross Millikan May 1 '11 at 0:23
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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
    
@cardinal: Yes, I agree. –  Ross Millikan May 1 '11 at 0:27
    
Thanks all for the help. –  Annoli May 1 '11 at 0:44

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