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For what kind of functions/or for which functions $h(a,b)$ is the below equation true :

assumption is $\lim_{ x\to\infty }{f(x)}=l_1$, $\lim_{ x\to\infty }{g(x)}=l_2.$

When is it true that $\lim_{ x\to\infty }{h(f(x),g(x))}=h(\lim_{ x\to\infty }{f(x)},\lim_{ x\to\infty }{g(x)}) = h(l_1, l_2)$ ?

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I think you meant to say $f(a,b)$ instead of $h(a,b)$, anyways your counterexample isn't one: $\lim_{x\to\infty} (1+1/x)^{1/x} = 1 = 1^0 = (\lim_{x\to\infty} 1 + 1/x)^{\lim_{x\to\infty} 1/x}$ –  Anthales Apr 11 '12 at 11:00
    
Thanks Anthales, I've made a few mistakes in my question. –  Qbik Apr 11 '12 at 12:03
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The answer is: when the function $h$ is continuous at $(l_1,l_2)$. –  Rahul Apr 11 '12 at 12:05

1 Answer 1

As observed by Rahul Narain, the implication $$ \lim_{x\to\infty} f(x)=l_1, \lim_{x\to\infty} g(x)=l_2 \implies \lim_{x\to\infty} h(f(x),g(x)) \tag1$$ holds if and only if the function $h$ is continuous at the point $(l_1,l_2)$. The same is true if $\infty$ is replaced by a number $c$ everywhere in (1).

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