limit involving $e$, ending up without $e$. Compute the limit
$$ \lim_{n \rightarrow \infty} \sqrt n \cdot \left[\left(1+\dfrac 1 {n+1}\right)^{n+1}-\left(1+\dfrac 1 {n}\right)^{n}\right]$$
we have a bit complicated solution using Mean value theorem. Looking for others

As Zhanxiong answered, Taylor expansion always works.
Consider, for large value of $x$, $$A(x)=\big(1+\frac 1x\big)^x$$ $$\log\big(A(x)\big)=x\log\big(1+\frac 1x\big)=x\big( \frac{1}{x}-\frac{1}{2 x^2}+\frac{1}{3 x^3}+\cdots\big)=1-\frac{1}{2 x}+\frac{1}{3 x^2}+\cdots$$ $$A(x)=e \, e^{-\frac{1}{2 x}+\frac{1}{3 x^2}}$$ So $$A(n+1)-A(n)\approx e\Big(e^{-\frac{1}{2 (n+1)}+\frac{1}{3 (n+1)^2}} -e^{-\frac{1}{2 n}+\frac{1}{3 n^2}}\Big)$$ Now, using , for small $y$, $e^y=1+y+\frac{y^2}2+\cdots$ $$A(n+1)-A(n)\approx e\Big(\frac{1}{2 n^2}-\frac{17}{12 n^3}+\cdots\Big)$$
