1. ## Simplifying a radical with two terms

Hi,

I sincerely hope this is the right forum. It's a limit problem, but the limit isn't my problem. Quick backstory if you care: I'm a 27 year old computer programmer who's trying to go through Calculus for Dummies this summer for my own education. So no, this isn't homework. I'm rapidly finding out I've forgotten a lot of basics and keep getting stuck on what I'm pretty sure is plain old algebra.

For instance, here's an example from the book:

$\lim_{x\to\infty}\frac{\sqrt{x^2+x}-x}{1}$

First step is to multiply by the conjugate of the numerator and simplify

$=\lim_{x\to\infty}\frac{\sqrt{x^2+x}-x}{1}\cdot\frac{\sqrt{x^2+x}+x}{\sqrt{x^2+x}+x}$
$=\lim_{x\to\infty}\frac{x^2+x-x^2}{\sqrt{x^2+x}+x}$

I'm good so far, but then it shows this:

$\lim_{x\to\infty}\left\frac{x}{x\left\sqrt{1+\frac {1}{x}}+1\right}\right$ (factor x out of the denominator)

The next step is just to cancel the from the numerator and denominator which I get and then do some substitution to solve the limit, but I'll save that for another post in another forum.

Anyways, I've been staring at, googling for, and whiteboard-experimenting that factoring step and I just don't follow it. It's been too long since I did any math, I guess. Can anybody explain it to me? I've subbed in some real numbers and it does work, but I don't know the rules that would let me jump from step 3 to 4.

2. ## Re: Simplifying a radical with two terms

$\frac{x}{\sqrt{x^2+x} + x} =$

$\frac{x}{\sqrt{x^2 \left(1 + \frac{1}{x} \right)} + x} =$

$\frac{x}{\sqrt{x^2} \cdot \sqrt{1 + \frac{1}{x}} + x} =$

$\frac{x}{x \sqrt{1 + \frac{1}{x}} + x} =$

$\frac{x}{x\left(\sqrt{1 + \frac{1}{x}} + 1\right)}$

3. ## Re: Simplifying a radical with two terms

Hi, and thanks, I finally get it! Although I must admit that step 2 still took a little bit of staring. I guess that's the kind of thing you just start to notice with experience? Hopefully...

Anyways, thanks again!