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?

Thank you for your help.

  • 2
    $\begingroup$ Hint: What is the limit of $(n+1)/(n-1)$ ? $\endgroup$ – Martin R Jan 23 '19 at 12:40
  • 2
    $\begingroup$ Because $(n-1)/(n+1) \not\to +\infty$, and you cannot use that significant well-known limit about $\mathrm e$. $\endgroup$ – xbh Jan 23 '19 at 12:41
  • $\begingroup$ limit of $(n+1)/(n-1)=1$, thank you! $\endgroup$ – J.Doe Jan 23 '19 at 12:41

I hope you solved the problem. Here is a fact that might help you.

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, $$\lim_{n\to\infty}(1+\frac{n-1}{n+1})^{\frac{n+1}{n-1}}= \lim_{t\to 1}(1+\frac{1}{t})^t=2$$


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