Answer
Horizontal asymptote:$\qquad y=5$
(graphed at desmos.com)
Work Step by Step
$\displaystyle \lim_{x\rightarrow\infty}(\frac{5x^{2}}{x^{2}+2})=\lim_{x\rightarrow\infty}\frac{5x^{2}\div x^{2}}{(x^{2}+2)\div x^{2}}=\lim_{x\rightarrow\infty}\frac{5}{1+2/x^{2}}=5$
$\displaystyle \lim_{x\rightarrow-\infty}(\frac{5x^{2}}{x^{2}+2})=5$
Horizontal asymptote:$\qquad y=5$
