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, $|x| + 6$ is never zero because the absolute value function $|x|$ is always non-negative, and adding 6 ensures the denominator is always positive. Therefore, there are no vertical asymptotes. For horizontal asymptotes, we examine the behavior as $x$ approaches infinity. The highest power of $x$ in both the numerator and denominator is $x^1$, so the horizontal asymptote is determined by the coefficients of the highest power terms, which gives $y = \frac{2}{1} = 2$. However, because the absolute value in the denominator can affect the sign depending on the direction from which $x$ approaches infinity, we also have a horizontal asymptote at $y = -2$.

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 $f ( x ) = \frac { 2 x - 8 } { | x | + 6 }$