Which limits involve computing square roots, Such as $\sqrt{n^2+6} -n$

Question

Please help me to computing this square limit. And if possible give me some website where have same tasks. Thank you! $\sqrt{n^2+6} -n$

Answer 1

The usual trick is to realize that

$$\sqrt{a} - \sqrt{b}$$

can be slightly simplified by multiplying by $$ \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}} $$ In your case, $a = n^2 + 6$ and $b = n^2$. Give it a try and see how it simplifies.

It relies on the idea that $(x + y) (x - y) = x^2 - y^2$, so if $x$ is written as a square-root, the right hand side does NOT have that square root.

Answer 2

Take a few terms of the binomial expansion:

$$\sqrt{n^2+6}=n+\frac3n+\mathcal O\left(\frac1{n^2}\right)$$

Thus,

$$\sqrt{n^2+6}-n=\frac3n+\mathcal O\left(\frac1{n^2}\right)$$