Originally Posted by

**Capes** So, the equation is:

$\displaystyle \lim (\sqrt{9x^2 + x} -3x) $

$\displaystyle x\rightarrow\infty$

Well, I first tried solving in the way i solved equations that didn't contain sqrt, dividing all the equation by the biggest power of x, in this case, dividing by $\displaystyle X$, getting

$\displaystyle \lim (\sqrt{9 + 1/x} -3) $

$\displaystyle x\rightarrow\infty$

And, by limit laws i would get $\displaystyle 3-3 = 0$, however, according to my book (which I trust more than myself) the answer is 1/6, so I tried doing in a different way:

$\displaystyle \lim (\sqrt{9x^2 + x} -3x)*(\sqrt{9x^2 + x} +3x)/(\sqrt{9x^2 + x} +3x) $

$\displaystyle x\rightarrow\infty$

and now getting :

$\displaystyle 1/(\sqrt{9 + 1/x} + 3)$, which is, $\displaystyle 1/6$.

So, my question is, why I couldn't do it the first way? Isn't it supposed to be the same thing? I mean, clearly not but, what rule am I breaking by doing it in the first way?

Thanks in advance