<s>[INST] Computing $ \lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right]$ 
$$
\lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right] \text { equals }\_\_\_\_
$$

I tried to expand the term in power using binomial theorem but still could not obtain the limit. 
 [/INST] One has
$$n - \frac{n}{e} \left( 1 + \frac{1}{n}\right)^n = n - \frac{n}{e} \exp \left( n \ln \left( 1 + \frac{1}{n}\right)\right) =n - \frac{n}{e} \exp \left( n \left(\frac{1}{n}- \frac{1}{2n^2} + o \left( \frac{1}{n^2}\right)\right)\right)  $$
so 
$$n - \frac{n}{e} \left( 1 + \frac{1}{n}\right)^n = n - \frac{n}{e} \exp \left( 1 -\frac{1}{2n} + o \left( \frac{1}{n}\right)\right) = n - n \left( 1 - \frac{1}{2n} + o \left( \frac{1}{n}\right)\right) $$
so
$$n - \frac{n}{e} \left( 1 + \frac{1}{n}\right)^n = \frac{1}{2} + o(1).$$
Therefore the limit is 
$$\frac{1}{2}$$
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