Find all horizontal and vertical asymptotes of the function $y ( x ) = \frac { 6 } { x - 6 }$ .
A) HA: $x - a x i s$
$\mathrm { VA } : x = 6$
B) $\mathrm { HA: } : y = 0$
$\mathrm { V A}$ : $x = - 6$
C) $\mathrm { HA }$ : none
$\mathrm { VA } : x = 6$
D) $\mathrm { HA }$ : none
$\mathrm { VA } : x = 1$
E) $\mathrm { HA } : y = 6$
$\mathrm { VA } : x = 6$

Question

Find all horizontal and vertical asymptotes of the function $y ( x ) = \frac { 6 } { x - 6 }$ .
A) HA: $x - a x i s$
$\mathrm { VA } : x = 6$ 
B) $\mathrm { HA: } : y = 0$
$\mathrm { V A}$ : $x = - 6$ 
C) $\mathrm { HA }$ : none
$\mathrm { VA } : x = 6$ 
D) $\mathrm { HA }$ : none
$\mathrm { VA } : x = 1$ 
E) $\mathrm { HA } : y = 6$
$\mathrm { VA } : x = 6$

Quizplus · Accepted Answer

To find the horizontal asymptote, we need to examine what happens to the function as x approaches positive or negative infinity.

As x approaches infinity, the function approaches zero. This is because as x gets larger and larger (in the positive direction), the denominator x - 6 also gets larger and larger, so the fraction 6/(x-6) gets smaller and smaller, and approaches zero as x approaches infinity.

Similarly, as x approaches negative infinity, the function also approaches zero.

Therefore, the horizontal asymptote is y=0.

To find the vertical asymptote, we need to look for values of x that make the denominator of the fraction zero. In this case, the denominator is x-6, so the only value of x that makes the denominator zero is x=6.

Therefore, the vertical asymptote is x=6.

The best choice is A, as it correctly identifies both the horizontal and vertical asymptotes.

Find All Horizontal and Vertical Asymptotes of the Function y(x)=6x−6y ( x ) = \frac { 6 } { x - 6 }y(x)=x−66​

Find All Horizontal and Vertical Asymptotes of the Function $y ( x ) = \frac { 6 } { x - 6 }$