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I need a (sketch of) proof (preferably by delta-epsilon) that:

1. If $\lim f(x) = L$, $0 < L < 1$ and $\lim g(x) = \infty$ then $\lim f(x)^{g(x)} = 0$

2. If $\lim f(x) = L$, $L > 1$ and $\lim g(x) = \infty$ then $\lim f(x)^{g(x)} = \infty$

3. If $\lim f(x) = L$, $L > 0$ and $\lim g(x) = \infty$ then $\lim g(x)^{f(x)} = 0$

All limits are of $x$ approaching an arbitrary number $a$.

Thank you!

Edit: I have that

$0<|x-a|

$0<|x-a|M=\log_{L+e} e$.

This gives $f(x)^{g(x)} < (L+e)^{g(x)}$.

Now I need $(L+e)^{g(x)} < (L+e)^{\log_{L+e} e} = e$, but I can't prove $L+e < 1$.

If I can, #2 can be proven with $\frac{1}{L}$ instead of $L$ also.

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    \infty${}{}{}{}$ – 2017-02-02
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    Please offer some idea of what you've tried on this, or what you know how to do. If you don't do this, people will be less inclined to help. – 2017-02-02

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