A function is said to have a horizontal asymptote if either the limit at infinity exists or the limit at negative infinity exists.
Show that each of the following functions has a horizontal asymptote by calculating the given limit.
$\displaystyle{ \lim_{x\to\infty}\frac{-4 x}{4+2 x}= }$
$\displaystyle{ \lim_{x\to-\infty}\frac{5 x-7}{x^3+7 x-12}= }$
$\displaystyle{ \lim_{x\to\infty}\frac{x^2-4 x -5}{13-10 x^2}= }$
$\displaystyle{ \lim_{x\to\infty}\frac{\sqrt{x^2+13 x}}{9-12 x}= }$
$\displaystyle{ \lim_{x\to-\infty}\frac{\sqrt{x^2+13 x}}{9-12 x}= }$