Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $C>1$, $f(x) < Cg(x)$.

share|improve this question
2  
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

1 Answer 1

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 $f(x)>g(x).$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.