Answer
Vertical asymptotes: none
Holes: none
Work Step by Step
Locating Vertical Asymptotes, (page 395) tells us that for $f(x)=\displaystyle \frac{p(x)}{q(x)}$,
if p(x) and q(x) have NO common factors,
and a is a zero of the denominator $q(x)$ ,
then the line $x=a$ is a vertical asymptote.
BUT, if they DO have common factors,
after REDUCING the form of the function's equation,
the number a may not cause the denominator to be zero any more.
in which case there will be a hole at x=a.
The point is that both p(x) and q(x) have to be factored,
and if we can, then we reduce the expression for f(x).
------------------
$r(x)=\displaystyle \frac{x}{x^{2}+3}$
$p(x)=x\qquad $... fully factored
$q(x)=x^{2}+3\qquad $... fully factored
Nothing to reduce....
$q(x)=0$
$x^{2}+3=0$
$x^{2}=-3$
... no real number has a negative square.
Since the denominator , q(x) has no zeros
r is defined for all real numbers,
meaning
Vertical asymptotes: none
Holes: none
