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Math Help - Simplifying a radical with two terms

  1. #1
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    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.

    Thanks in advance!
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  2. #2
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    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)}
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  3. #3
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    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!
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