# How would one compute lim_(delta => 0, k => infty)(1+delta)^(ak), where a is some positive constant?

How would one compute $\underset{\delta \to 0,k\to \mathrm{\infty }}{lim}\left(1+\delta {\right)}^{ak}$, where a is some positive constant?
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Reese King
It’s undefined, because the limit depends entirely on how $k\to \mathrm{\infty }$ and $\delta \to 0$. For example:
$\underset{k\to \mathrm{\infty }}{lim}\underset{\delta \to 0}{lim}\left(1+\delta {\right)}^{ak}=\underset{k\to \mathrm{\infty }}{lim}{1}^{ak}=1$
$\underset{\delta \to {0}^{+}}{lim}\underset{k\to \mathrm{\infty }}{lim}\left(1+\delta {\right)}^{ak}=\underset{\delta \to {0}^{+}}{lim}\mathrm{\infty }=\mathrm{\infty }$
$\underset{\delta \to {0}^{-}}{lim}\underset{k\to \mathrm{\infty }}{lim}\left(1+\delta {\right)}^{ak}=\underset{\delta \to {0}^{-}}{lim}0=0$
$\underset{\delta \to {0}^{+}}{lim}\left(1+\delta {\right)}^{a/\delta }=\underset{k\to \mathrm{\infty }}{lim}{\left(1+\frac{1}{bk}\right)}^{ak}={e}^{a/b}$