Question

Find the value of the following limit:
$\underset{x\to \infty }{lim}(x-\sqrt{x^2+5x})$

Answer 1

$\underset{x\to \infty }{lim}x-\sqrt{x^2-5x}$

Using Rationalization here:

$=\underset{x\to \infty }{lim}\frac{x-\sqrt{x^2+5x}}{1}\times \frac{(x+\sqrt{x^2+5x})}{(x+\sqrt{x^2+5x})}$

$=\underset{x\to \infty }{lim}\frac{x^2-(x^2+5x)}{x+\sqrt{x^2+5x})}=\underset{x\to \infty }{lim}\frac{x^2-x^2-5x}{(x+\sqrt{x^2+5x})}$

$=\underset{x\to \infty }{lim}\frac{-5x}{x+\sqrt{x^2+5x}}$

$=\underset{x\to \infty }{lim}\frac{-5}{(1+\sqrt{1+\frac{5}{x}})}$

$=\frac{-5}{(1+\sqrt{1+\frac{5}{\infty }})}$

$=\frac{-5}{2}$.