Limit $\lim_{n\to\infty} n^2\left(\sqrt{1+\frac{1}{n}}+\sqrt{1-\frac{1}{n}}-2\right)$

Question

Greetings I am trying to solve $$\lim_{n\to\infty} n^2\left(\sqrt{1+\frac{1}{n}}+\sqrt{1-\frac{1}{n}}-2\right)$$ Using binomial series is pretty easy: $$\lim_{n\to\infty}n^2\left(1+\frac{1}{2n}-\frac{1}{8n^2}+\mathcal{O}\left(\frac{1}{n^3}\right)+1-\frac{1}{2n}-\frac{1}{8n^2}+\mathcal{O}\left(\frac{1}{n^3}\right)-2\right)=\lim_{n\to\infty}n^2\left(-\frac{1}{8n^2}+\mathcal{O}\left(\frac{1}{n^3}\right)-\frac{1}{8n^2}+\mathcal{O}\left(\frac{1}{n^3}\right)\right)=-\frac{1}{4}$$ The problem is that I need to solve this using only highschool tools, but I cant seem too take it down. My other try was to use L'Hospital rule but I feel like it just complicate things. Maybe there is even an elegant way, could you give me some help with this?

Answer 1

Hint: multiplying numerator and denominator by $\sqrt{1+1/n}+\sqrt{1-1/n}+2$ we get $$2 n^2 \frac{\sqrt{1-1/n^2}-1}{\sqrt{1+1/n}+\sqrt{1-1/n}+2}$$ and then do the same with $$\sqrt{1-1/n^2}+1$$ you will get

$$\frac{n^2(2(\sqrt{1-1/n^2}-1))(\sqrt{1-1/n^2}+1)}{(\sqrt{1+1/n}+\sqrt{1-1/n}+2)(\sqrt{1-1/n^2}+1)}$$

Answer 2

Making $\delta = \frac 1n$ You can arrange it as

$$ \lim_{\delta\to 0}\left(\frac{\frac{\sqrt{1+\delta}-1}{\delta}-\frac{\sqrt{1-\delta}-1}{\delta}}{\delta}\right) = \left(\frac{d^2}{dx^2}\sqrt x\right)_{x=1} = -\frac 14 $$

Answer 3

Here's a slightly different approach using multiplication by conjugates.

For notational simplicity, let $u=1/n$, so that $u\to0$ as $n\to\infty$. Then

$$n^2\left(\sqrt{1+{1\over n}}+\sqrt{1-{1\over n}}-2\right)={\sqrt{1+u}-1\over u^2}+{\sqrt{1-u}-1\over u^2}\\ ={1\over u(\sqrt{1+u}+1)}-{1\over u(\sqrt{1-u}+1)}\\={\sqrt{1-u}-\sqrt{1+u}\over u(\sqrt{1+u}+1)(\sqrt{1-u}+1)}\\={-2\over(\sqrt{1+u}+1)(\sqrt{1-u}+1)(\sqrt{1-u}+\sqrt{1+u})}\to{-2\over(1+1)(1+1)(1+1)}=-{1\over4}$$