Determine the equations of any horizontal and vertical asymptotes of
$$f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 16 } .$$
A) horizontal: $y = 4$; vertical: $x = - 4$
B) horizontal: $x = 1$; vertical: none
C) horizontal: $y = - 4$; vertical: $x = 1$
D) horizontal: $y = 1$; vertical: none
E) horizontal: none; vertical: none

Question

Quizplus · Accepted Answer

Horizontal asymptotes are found by examining the behavior of a function as $x$ approaches infinity. For $f(x) = \frac{x^2}{x^2 + 16}$, as $x$ approaches infinity, the value of $f(x)$ approaches $1$, because the degrees of the polynomials in the numerator and denominator are the same, and their leading coefficients are both 1. Therefore, the horizontal asymptote is $y = 1$. There are no vertical asymptotes because the denominator $x^2 + 16$ cannot be zero for any real value of $x$, as the sum of a square and a positive number is always positive.

Determine the Equations of Any Horizontal and Vertical Asymptotes Of f