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x- \sqrt{ x^{2} +x} = \frac{(x- \sqrt{x^{2}+x})(x+ \sqrt{x^{2}+x }) }{x+ \sqrt{x^{2}+x}}
= \frac{x^2-x^2-x}{x+ \sqrt{x^2+x} }= \frac{-x}{x+ \sqrt{x^2+x}}= \frac{-1}{1+ \sqrt{ \frac{1}{x} +1} }
now take limit
u will get  \frac{-1}{2}
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