# Useful theorems/properties for evaluating $\lim \limits_{x \to +\infty}\sqrt{x}(\sqrt{x+1}-\sqrt{x-1})$

I have been trying to solve the following question from my Mathematical Analysis course in university:

$$\lim \limits_{x \to +\infty}\sqrt{x}(\sqrt{x+1}-\sqrt{x-1})$$

I am aware that the answer is 1, but I am not entirely sure why.

Substituting positive infinity into the equation, if I am not terribly mistaken, gives $$\infty-\infty$$, which is indeterminate. Anyhow, I am not aware of any properties or theorems that could get rid of this issue, and Landau symbols don't really seem to help. I have tried as well to rewrite the problem as $$\lim \limits_{x \to +\infty}\frac{\sqrt{x+1}-\sqrt{x-1}}{\frac{1}{\sqrt{x}}}$$

to use L'Hopital's theorem, which does not get rid of the indeterminate form. Help would be appreciated. Thanks in advance!

• $\sqrt a - \sqrt b = \frac{a-b}{\sqrt a + \sqrt b}$ helps quite often ... Commented Jun 9, 2022 at 20:06

In fact Landau symbols do work very well.

$$\sqrt{1\pm u}=1\pm\frac 12 u+o(u)\$$ for $$\ u\to 0$$

So factor out $$x$$ to get:

\begin{align}\require{cancel}x\left(\sqrt{1+\frac 1x}-\sqrt{1-\frac 1x}\right)&=x\Big((\cancel{1}+\frac 1{2x}+o(\frac 1x))-(\cancel{1}-\frac 1{2x}+o(\frac 1x))\Big)\\&=x\Big(\frac1x+o(\frac 1x)\Big)\\&=1+o(1)\\&\to 1\end{align}

But you can as well use the conjugated quantity method as always when difference of square roots is used:

$$\sqrt{x+1}-\sqrt{x-1}=\dfrac{(x+1)-(x-1)}{\sqrt{x+1}+\sqrt{x-1}}=\dfrac{2}{\sqrt{x+1}+\sqrt{x-1}}\sim \dfrac 2{2\sqrt{x}}\sim\dfrac 1{\sqrt{x}}$$

And multiplied by $$\sqrt{x}$$ this indeed has limit $$1$$.

Note that if $$f(x)=\sqrt{x(x-1)}$$ this is equivalent to finding $$\lim_{x\to\infty} f(x+1)-f(x)$$ Note that for $$x>1$$, $$\sqrt{x^2-x}<\sqrt{x^2-x+.25}=x-.5$$. Also it's not hard to prove that for any $$\epsilon<0$$, there exists $$\delta$$ such that $$\forall x>\delta$$, $$\sqrt{x^2-x}>x-.5-\epsilon$$. In particular this value of $$\delta$$ is $$\delta=\frac{(.5+\epsilon)^2}{2\epsilon}$$. So $$\forall \epsilon>0$$, we know that $$\forall x>\delta$$, $$1-\epsilon. So it follows that $$\lim_{x\to\infty} f(x+1)-f(x)=1$$.