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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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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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    @Annoli, please (formally) accept @Ross' answer if you feel it answers your question. Cheers.2011-05-01