limit of functions $f(x)>g(x)$, what can we conclude about their limits

Asked Mar 01 '21 at 20:08

Active Mar 01 '21 at 20:35

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Let $f$ and $g$ be two functions defined on $S$ and $c$ be a cluster point of $S$. Suppose $f(x)\to l_1$ and $g(x)\to l_2$ as $x\to c$. If $f(x)>g(x)$ for any $x\in S\setminus{\{c\}}$,can we conclude $l_1 > l_2$?

I think not and one of the counter examples I can think of is $x^2$ and $2x^2$, $c=0$?

edited Mar 01 '21 at 20:35

jjagmath

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asked Mar 01 '21 at 20:08

Oneoneonesong

1

As a rule of thumb, taking limits turns strict inequalities to non-strict – shortmanikos Mar 01 '21 at 20:13
Please, learn to use mathjax. Your questions will be better received in this site if you show more effort writing them. – jjagmath Mar 01 '21 at 20:38
Yes, your counterexample is right. – jjagmath Mar 01 '21 at 20:38

limit of functions $f(x)>g(x)$, what can we conclude about their limits

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