Question

Evaluate $\underset{x\to \infty }{lim}\sqrt{x^2+x+}-\sqrt{x^2+1}$

Answer 1

$\begin{array}{c}{lim}_{x\to \infty }\sqrt{x^2+x}-\sqrt{x^2+1}\\ {lim}_{x\to \infty }x\sqrt{1+\frac{x}{x^2}}-x\sqrt{1+\frac{1}{x^2}}\\ {lim}_{x\to \infty }\left(1+\frac{1}{2x}-1-\frac{1}{2x^2}\right)\\ {lim}_{x\to \infty }\frac{1}{2x}-\frac{1}{2x^2}\\ \frac{1}{2}=0\end{array}$