Limit of $0^0$ (Indeterminate form)

Your claim is wrong. You can choose any nonnegative number as the limit: $$\lim_{x\to0^+}\left(e^{-1/x}\right)^{ax} = e^{-a}.$$

This example is from the Wikipedia article on $0^0$, as @PrasunBiswas suggested. There are other examples there showing a limit of infinity, etc.

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Limits

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