Answer
Vertical asymptote: None
Horizontal asymptote: $y=0$
Work Step by Step
$r(x)=\dfrac{3x+1}{4x^{2}+1}$
Vertical asymptotes
A rational function has vertical asymptotes where the function is undefined, that is, where the denominator is zero.
Set the denominator equal to $0$ and solve for $x$:
$4x^{2}+1=0$
$4x^{2}=-1$
$x^{2}=-\dfrac{1}{4}$
$x=\sqrt{-\dfrac{1}{4}}$
$x=\pm\dfrac{1}{2}i$
Since the answer is a complex number, this function doesn't have vertical asymptotes.
Horizontal asymptote
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote of this function is $y=0$
