Determine the equations of any horizontal and vertical asymptotes of $$f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 4 }$$ .
A)horizontal: y = 2;vertical: x = -2
B)horizontal: x = 1;vertical: none
C)horizontal: none;vertical: none
D)horizontal: y = 1;vertical: none
E)horizontal: y = -2;vertical: x = 1

Question

Quizplus · Accepted Answer

For horizontal asymptotes, we look at the behavior of the function as $x$ approaches infinity. The degree of the numerator and denominator are the same, so we divide the leading coefficients: $1/1 = 1$, giving us a horizontal asymptote at $y = 1$. For vertical asymptotes, we look for values of $x$ that make the denominator zero. However, $x^2 + 4 = 0$ has no real solutions (since $x^2$ is always non-negative, $x^2 + 4$ is always greater than zero), so there are no vertical asymptotes.

Determine the Equations of Any Horizontal and Vertical Asymptotes Of f