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Here is my question:

$\displaystyle\lim_{n\rightarrow \infty}n^2[(1+\frac{1}{1+n})^{n+1}-(1+\frac{1}{n})^{n}]=?$

Any hints will be fine. Thank you!

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As simple as using Limit:

Clear[n];
Limit[n^2 ((1 + 1/(1 + n))^(n + 1) - (1 + 1/n)^n), n -> ∞]
(*E/2*)
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