Evaluate the limit of  $\lim_{x\to\infty}\frac{x - \sqrt{x^2+5x+2}}{x-\sqrt{x^2+0.5x + 1}}.$ $$\lim_{x\to\infty}\frac{x - \sqrt{x^2+5x+2}}{x-\sqrt{x^2+0.5x + 1}}.$$
Can't find a means to resolve. The answer is $10$ by graphing.  

Divide through by $x, \lim \limits_{x\to\infty}\frac{x - \sqrt{x^2+5x+2}}{x-\sqrt{x^2+0.5x + 1}}=\lim \limits_{x\to\infty}\frac{1 - \sqrt{1+5/x+2/x^2}}{1-\sqrt{1+0.5/x + 1/x^2}}$ then use the expansion around $x=\infty \lim  \limits_{x\to\infty}\frac{1 - \sqrt{1+5/x+2/x^2}}{1-\sqrt{1+0.5/x + 1/x^2}} \approx \frac{1-[1+5/(2x)]}{1-[1+1/(4x)]}=10$ where the $\approx$ means here that the first orders are the same.
