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I'd like to know how this might be proven, or why it's true:

$\begin{aligned} \lim\limits_{n\to\infty}{\left( 1+\frac{1}{n^k}\right)^n} = \infty, \text{when } k<1\\ = e, \text{ when } k=1\\ =1, \text{ when } k>1 \end{aligned}$

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It's not true --- have another look at your source, you may have made copying errors. –  Gerry Myerson Feb 12 at 6:16
    
I've corrected it now... I think. –  user64325 Feb 12 at 6:33

1 Answer 1

Hint

For the second case, I let you going back to the definition of $e$.

For the other cases, for large values of $n$, $1/n^k$ is small. Remember that if $x$ is small, $(1+x)^n$ can be approximated by $(1+n x)$. So, replace $x$ by $1/n^k$, simplify and ... conclude.

I am sure you can take from here.

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