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Suppose $ f $ is a positive function. If $ lim_{x\to a} f(x) = 0 $ and $ lim_{x\to a} g(x) = \infty $, show that

$$ \displaystyle \lim_{x\to a} [f(x)]^{g(x)} = 0 $$

This shows that $ 0^{\infty} $ is not an indeterminate form.

$\lim _{x \rightarrow a} y=\lim _{x \rightarrow a} f(x)^{g(x)}=\lim _{x \rightarrow a} e^{\ln y}=\lim _{h \rightarrow-\infty} e^{h}=0$

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Campbell University

University of Nottingham

Idaho State University

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