Question

$\lim_{x \to \infty} [x- \sqrt{ x^{2} +x} ]$

Answer 1

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