$=\lim\limits_{x\to -\infty}\dfrac{x^2-(x^2+x+1)}{x-\sqrt{x^2+x+1}}$
$=\lim\limits_{x\to -\infty}\dfrac{-x-1}{x+x.\sqrt{1+\dfrac{1}{x}+\dfrac{1}{x^2}}}$
$=\lim\limits_{x\to -\infty}\dfrac{-1-\dfrac{1}{x}}{1+\sqrt{1+\dfrac{1}{x}+\dfrac{1}{x^2}}}$
$=\dfrac{-1}{1+1}$
$=\dfrac{-1}{2}$