$\displaystyle y = x-4/(x^2+2)$
for the graph of
$\displaystyle y = \frac{x-4}{x^2+2}$
for horizontal asymptote, divide numerator and denominator with highest power $\displaystyle (x^2)$ and take $\displaystyle x\rightarrow \infty$
horizontal asymptote is x-axis (equation of x-axis is y=0)
there is no vertical asymptote.
$\displaystyle as\;\; x \rightarrow \infty, \;\;y \rightarrow 0$
$\displaystyle as\;\; x \rightarrow -\infty, \;\;y \rightarrow 0$
also, points (0, -2) and (4,0) lie on the graph.
Please see graph