Suppose that $f(x)^{g(x)},f(x),g(x)$ each converges, does the statement $\lim_{x\to\infty}f(x)^{g(x)}=\left[\lim_{x\to\infty}f(x)\right]^{\lim_{x\to\infty}g(x)}$ true?
Proof or disproof this statement on limit
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calculus
analysis
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0By converges do you mean that limit at infinity exists? – 2017-02-23
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0Yes. Limit at infinity is finite. – 2017-02-23
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4It is important to add that the limit of base $f(x) $ is positive. – 2017-02-23
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0If $f(x)$ converges to a positive value as $x \to \infty$, then you can just use continuity of the function from $(0,\infty)^2 \to \Bbb R$ given by $(x,y) \mapsto x^y = e^{y \log x}$. – 2017-02-23
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2To prove this just take logs and apply usual limit rules and the fact that $\log$ is continuous. – 2017-02-23
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2If the limit of $f$ is not positive then we may have negative values of $f$ and then the general power $f^{g} $ may not be a real number. – 2017-02-23
1 Answers
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If $a:=\lim_{x \to \infty}f(x)>0$ and $b:=\lim_{x \to \infty}g(x)$ , then
$f(x)^{g(x)}=e^{g(x)\ln (f(x)} \to e^{b\ln a} =a^b$ for $x \to \infty$, since $\exp$ and $ \ln$ are continuous.