<s>[INST] 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.
 [/INST] Note that 
$$ \sqrt{x^2 - x + 2} = \sqrt{x^2 -x + \frac{1}{4} + 1.75} = \sqrt{(x - \frac{1}{2})^2 + 1.75}$$.
What is the limit of $ \sqrt{(x - \frac{1}{2})^2 + 1.75} -  \sqrt{(x - \frac{1}{2})^2}$?
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