Thank you all again for your help. I've been studying with the L'hopital rule and for the most part haven't had any difficulty with it except for this one question. You guys have helped tremendously with this self-study course thus far.

It may be more of an algebra issue for me than anything else, but here's the question:

Use L'Hopital's rule to evaluate $\displaystyle % MathType!MTEF!2!1!+-

% feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuL wBLn

% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqt ubsr

% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9

% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x

% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaac i

% GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHEisPaeqa aOGa

% amiEaiabgkHiTmaakaaabaGaamiEamaaCaaaleqabaGaaGOmaa aaki

% abgUcaRiaadIhaaSqabaaaaa!4335!

$$\mathop {\lim }\limits_{x \to \infty } x - \sqrt {{x^2} + x} $$$

I am having difficulty getting this into a quotient where L'Hopital applies. For that reason I have two questions about this problem:

1) Is $\displaystyle % MathType!MTEF!2!1!+-

% feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuL wBLn

% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqt ubsr

% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9

% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x

% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaac q

% GHEisPcqGHsislcqGHEisPaeaacqGHEisPaaaaaa!3B43!

$${{\infty - \infty } \over \infty }$$$ an indeterminate form? To me it would be undefined because of the $\displaystyle \infty - \infty$, which isn't quite the same as being in an indeterminate form for L'hopital as far as I know (which I'm more than willing to be wrong about).

2) What would be the proper way to set this up for L'hopital?

Thank you again.