Determine the equations of any horizontal and vertical asymptotes of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 9 }$ .
A)horizontal: $y = 3$ ; vertical: $x = - 3$
B)horizontal: $x = 1$ ; vertical: none
C)horizontal: $y = - 3$ ; vertical: $x = 1$
D)horizontal: $y = 1$ ; vertical: none
E)horizontal: none; vertical: none

Question

Quizplus · Accepted Answer

The horizontal asymptote of a rational function can be found by comparing the degrees of the numerator and denominator. Since the degrees are the same, the horizontal asymptote is $y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} = \frac{1}{1} = 1$. There are no vertical asymptotes because the denominator $x^2 + 9$ cannot be zero for any real value of $x$, as $x^2 + 9 > 0$ for all $x$.

Determine the Equations of Any Horizontal and Vertical Asymptotes Of f