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If $f$ and $g$ are continuous, non-negative functions on real line such that $f(x)>g(x)$ for all $x\in\mathbb{R}$. Can you find an example of $f$ and $g$ such that for all $C>1$, $f(x) < Cg(x)$.

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Presumably, you meant "for each $C>1$, $f(x)<Cg(x)$ for some $x\in\Bbb R$". In this case, consider a function with a horizontal asymptote $y=a$ with $a> 0$ whose graph lies above the asymptote. – David Mitra Dec 20 '12 at 19:44

First note that from $g(x)\ge 0$ and $f(x)>g(x)$ we have $f(x)>0$ for all $x$.

If for some $x$ we had $g(x)=0$ then the inequality $f(x)<Cg(x)$ would contradict $f(x)>0.$ So we may assume $g(x)>0$ for all $x$.

Now take $C=1+\frac{1}{n}$, then for fixed $x$, from $f(x)<(1+\frac{1}{n})g(x)$ it follows, on letting $n \to \infty$, that $f(x) \le g(x)$, contradicting the requirement that $f(x)>g(x).$

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