what is the limit as $x$ approaches infinity of $x-\sqrt{x^2-x+2}$ $$
\lim_{x \to \infty} \left( x - \sqrt{x^2 - x +2 } \right)
$$
I've tried rationalizing the expression but after repeated applications of L'Hospital's rule, it doesn't feel like I'm getting anywhere.

Yes rationalize is  a good idea, indeed we obtain
$$x - \sqrt{x^2 - x +2 }=x - \sqrt{x^2 - x +2 }\cdot\frac{x + \sqrt{x^2 - x +2 }}{x + \sqrt{x^2 + x +2 }}=\frac{x^2-x^2+x-2}{x+ \sqrt{x^2 + x +2 }}$$
$$=\frac{x-2}{x+ \sqrt{x^2 + x +2 }}=\frac x x\frac{1-\frac 2x}{1+ \sqrt{1 + \frac1x +\frac2{x^2} }}\to \frac12$$
