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Find: $$\lim_{n \rightarrow \infty}\frac{1}{\left(\frac{n+1}{n}\right)^{n+1}-\frac{n+1}{n}}$$

Solution

\begin{aligned} \lim_{n \rightarrow \infty}\frac{1}{\left(\frac{n+1}{n}\right)^{n+1}-\frac{n+1}{n}}=&\lim_{n \rightarrow \infty}\left[\left(\frac{n+1}{n}\right)^{n+1}-\frac{n+1}{n}\right]^{-1}\\\\ =&\lim_{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)^{n+1}-\left(1+\frac{1}{n}\right)\right]^{-1}\\\\ =&\lim_{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)^{n} \left(1+\frac{1}{n}\right)-\left(1+\frac{1}{n}\right)\right]^{-1}\\\\ =&\frac{1}{e-1} \end{aligned}
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