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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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