How is this equation true?

Question

I was looking through my notes when I stumbled upon this eqation:

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

I was trying to make sense of it, but I didnt succeed.

Can you give me a hint or tell me how this tranformation is called ?

Answer 1

Multiply $\sqrt{n^2+n}+n$ top and bottom. In other words, we have:

$$\sqrt{n^2+n}-n=\frac{(\sqrt{n^2+n}-n)\times(\sqrt{n^2+n}+n)}{\sqrt{n^2+n}+n}$$

I think the rest you can figure it out on your own. We usually call such a method as 'rationalizing' the surd.

Answer 2

If you must find the limit of

$\sqrt{n^2 +n} -n ;\;n\to\infty$

You get an indeterminate form. Thus you multiply by $\sqrt{n^2 +n} +n $ the numerator and the denominator

$\dfrac{(n^2 +n) - n^2}{\sqrt{n^2 +n} +n}=\dfrac{n}{\sqrt{n^2 +n} +n}=\dfrac{n}{n\sqrt{1 +\frac{1}{n}} +n}=\dfrac{n}{n\left(\sqrt{1 +\frac{1}{n}} +1\right)}=\dfrac{1}{\sqrt{1 +\frac{1}{n}} +1}=\dfrac{1}{2}$

Hope this helps to recall the reason why you wrote down this note :)