# Compute $\lim_{n \to \infty}(1+\frac{n-1}{n+1})^{\frac{n+1}{n-1}}$

Compute $$\lim_{n \to \infty}(1+\frac{n-1}{n+1})^{\frac{n+1}{n-1}}$$

I did: $$\lim_{n\to \infty}(1+\frac{1}{\frac{n+1}{n-1}})^{\frac{n+1}{n-1}}=e$$.

Why is this incorrect?

• Hint: What is the limit of $(n+1)/(n-1)$ ? – Martin R Jan 23 '19 at 12:40
• Because $(n-1)/(n+1) \not\to +\infty$, and you cannot use that significant well-known limit about $\mathrm e$. – xbh Jan 23 '19 at 12:41
• limit of $(n+1)/(n-1)=1$, thank you! – J.Doe Jan 23 '19 at 12:41
Put, $$t=\frac{n+1}{n-1}$$. Here as $$n\to \infty$$, $$t=\frac{n+1}{n-1}=\frac{1+\frac{1}{n}}{1-\frac{1}{n}}\to 1$$. Putting all these we get, $$\lim_{n\to\infty}(1+\frac{n-1}{n+1})^{\frac{n+1}{n-1}}= \lim_{t\to 1}(1+\frac{1}{t})^t=2$$