Answer
Yes, it can.
(See below for reasons.)
Work Step by Step
If $g(x)$ is never zero, $\displaystyle \frac{f(x)}{g(x)}$ is defined for all real numbers, and there is no vertical asymptote.
But, f and g can have the same degree, in which case $\displaystyle \frac{f(x)}{g(x)}$ has a horizontal asymptote.
Furthermore, an oblique asymptote is possible, if we can write
$\displaystyle \frac{f(x)}{g(x)}=(ax+b)+\frac{1}{g(x)}$
