Ask Me Help Desk - Proving a limit

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proving a limit

I need help proving lim as n -> infinity of (1 + a/n)^n = e^a

We can build off the classic $\lim_{t\to\0}(1+t)^{\frac{1}{t}}=e$

We have:

$\lim_{n\to\infty}(1+\frac{a}{n})^{n}$

Let $t=\frac{a}{n}, \;\ n=\frac{a}{t}$

Now, as ${n\to\infty}$ , then ${t\to\0}$

$\lim_{t\to\0}(1+t)^{\frac{a}{t}}$

Factor out the a:

$\lim_{t\to\0}\left[(1+t)^{\frac{1}{t}}\right]^{a}$

As we can now see, the inner limit is e and we have $e^{a}$

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