as in the topic I find it difficult to calculate following limit:$\lim_{n\rightarrow\infty}(\frac{n-1}{n+1})^{b_{n}}$where $b_{n}=(\sqrt{n+1}-\sqrt{n-1})^{-2}$. Any ideas and hints would be appreciated; thanks in advance!
Problematic limit calculation
2 Answers
This limit is of $1^\infty$. The standard way to solve them is by taking logarithms.
$P_n = (\frac{n-1}{n+1})^{b_n}$
Let $p_n = \log(P_n) = b_n \log(\frac{n-1}{n+1})$
By the standard expansion of log $\log(\frac{n-1}{n+1}) = \log(1-\frac{2}{n+1}) \approx -\frac{2}{n+1} + O(\frac{1}{n^2})$
By rationalization, $b_n$ can be written as
$b_n = \frac{1}{(\sqrt{n+1}-\sqrt{n-1})^2} = \frac{(\sqrt{n+1}+\sqrt{n-1})^2}{({(n+1)}-{(n-1)})^2} = \frac{2(n+\sqrt{n^2-1})}{4}$
So,$p_n = \frac{2(n+\sqrt{n^2-1})}{4} (-\frac{2}{n+1}+O(n^{-2}))$
$p_n = -\frac{n+\sqrt{n^2-1}}{n+1} + O(n^{-1})$
Taking the limit
$\lim_{n \to \infty} p_n = -2$
So,
$\lim_{n \to \infty} P_n = \exp(-2)$
Hint: Rewrite this as $\left(1-\frac{a_n}n\right)^{nb_n}$ for some $(a_n)$ and $(b_n)$ such that $a_n\to a$ and $b_n\to b$. Then think about the behaviour of $\left(1-\frac{a}n\right)^{nb}$ when $n\to\infty$.