Question about limits with variable on exponent

Question

So I have to find the following limit $$\lim_{n\to\infty}\left(1+\frac{2}{n}\right)^{1/n}.$$I said that this is $$\lim_{n\to\infty}\left[\left(1+\frac{2}{n}\right)^n\right]^{1/n^2}=\left(e^2\right)^{\lim_{n\to\infty}1/n^2}=1.$$Now I know that the final answer is correct, but my method seems to me to be wrong. Can I seperate a limit in the way I did - i.e. is my working correct? Is there any other more elegant way to find the limit?

Thanks in advance.

I think your procedure has no defect at all. – A.D May 18 '13 at 14:06 — A.D, May 18 '13 at 14:06
I'm tempted to use the replacement $h=\frac1n$ – lab bhattacharjee May 18 '13 at 14:37 — lab bhattacharjee, May 18 '13 at 14:37

score 3 · Accepted Answer · answered May 18 '13 at 14:39

3

You don't actually need to go through this all: $$\lim_{n\rightarrow\infty} 1+\frac{2}{n}=1$$ and $$\lim_{n\rightarrow\infty} \frac{1}{n}=0$$ therefore $$\lim_{n\rightarrow\infty} \left(1+\frac{2}{n}\right)^{1/n}=1^0=1$$

answered May 18 '13 at 14:39

Peter Košinár

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Question about limits with variable on exponent

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