I've shown $\lim_{n\to\infty}\sqrt[n]{n}=e$, but it should be $1$. Where's the mistake? $$\lim_{n\to \infty} \sqrt[n] n = \lim_{n\to \infty} n^{\frac{1}{n}} = \lim_{n \to \infty} \{(1+(n-1))^{\frac{1}{n-1}}\}{^{(n-1)\frac{1}{n}}} = \lim_{n\to \infty} e^{\frac{n-1}{n}} = e$$
But this clearly isn't true as the actual limit is $1$.
Where did I go wrong? 

This is a wrong answer because you do $(1 + \infty)^{(1/ \infty)}$ is $e$ but Euler defined this limit by $(1+(\text{something that go to zero}))^{(1/\text{the something in the base that go to zero})} = e$, and that for why your answer wrong.
