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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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we know the first thing we can do is we can rewrite it in terms of natural log of why is G of axe times Ellen of f of axe to indicate the same thing? Because now we can more easily take the limit as extra purchase. A G evac stems the natural log of after wax, and we know this is equivalent to negative infinity. Now, given this we know the limit is ex purchase A of aftereffects of natural of affects is negative. Infinity and we know the woman is expertise have affects zero. Therefore, given this what we know is that we can rewrite this as the limit as Expro. Ches eh Eat the natural log of why we know this is the limit as H approaches negative infinity. You see, EJ, we know you did the negative. Infinity is simply going to be zero

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