Identify any horizontal and vertical asymptotes of the function below.
$$f ( x ) = \frac { 2 x - 8 } { | x | + 6 }$$
A) vertical asymptotes: $x = - 2$ and $x = 2$; horizontal asymptotes: $y = - 6$ and $y = 6$
B) vertical asymptotes: $x = - 2$ and $x = 2$; horizontal asymptotes: none
C) vertical asymptotes: $x = - 6$ and $x = 6$; horizontal asymptotes: none
D) vertical asymptotes: none; horizontal asymptotes: $y = - 2$ and $y = 2$
E) vertical asymptotes: $x = - 6$ and $x = 6$; horizontal asymptotes: $y = - 2$ and $y = 2$

Question

Quizplus · Accepted Answer

Vertical asymptotes occur where the denominator is zero. However, since $|x| + 6$ is never zero for any real value of $x$ (the absolute value function $|x|$ is always non-negative, and adding 6 ensures the denominator is always positive), there are no vertical asymptotes. Horizontal asymptotes are determined by the behavior of the function as $x$ approaches infinity or negative infinity. In this case, as $x$ approaches infinity or negative infinity, the function $f(x) = \frac{2x - 8}{|x| + 6}$ approaches $y = 2$ or $y = -2$, respectively, because the leading terms in the numerator and denominator are $2x$ and $|x|$, and their coefficients determine the horizontal asymptotes.

Identify Any Horizontal and Vertical Asymptotes of the Function Below f(x)=2x−8∣x∣+6f ( x ) = \frac { 2 x - 8 } { | x | + 6 }f(x)=∣x∣+62x−8​

Identify Any Horizontal and Vertical Asymptotes of the Function Below $f ( x ) = \frac { 2 x - 8 } { | x | + 6 }$