Question

using L'hopital's rule evaluate lim x-->infinity (x-sqrt(x^2 +x)

Answer 1

-1/2

Answer 2

do we have to use L`hopital rules ?

Answer 3

\[\lim_{x \rightarrow \infty}(x-\sqrt{x^2+x})\]
\[\lim_{x \rightarrow \infty}(x-\sqrt{x^2+x})*\frac{(x+\sqrt{x^2+x})}{(x+\sqrt{x^2+x})}\]
\[\lim_{x \rightarrow \infty}\frac{-x}{(x+\sqrt{x^2+x})}\]
\[\lim_{x \rightarrow \infty}\frac{-x}{x(1+\sqrt{1+\frac1{x^2}})}\]
\[\lim_{x \rightarrow \infty}\frac{-1}{(1+\sqrt{1+\frac1{x^2}})}=\frac{-1}2\]