How do you find vertical, horizontal and oblique asymptotes for #(x^2 - 9)/(3x-6)#?

Question

Theodore Amidon · Accepted Answer

Vertical Asymptote: #color(blue)(x = 2)#

Horizontal Asymptote: None

Equation of the Slant/Oblique Asymptote: #color(blue)(y = x/3+2/3)#

Given:

#color(red)(y = f(x) = (x^2-9)/(3x-6))#

#color(green)(Step.1):#

To find the Vertical Asymptote:

a. Factor where possible

b. Cancel common factors, if any

c. Set Denominator = 0

We will start following the steps:

Consider:

#color(red)(y = f(x) = (x^2-9)/(3x-6))#

We will factor where possible:

#y = f(x) = [(x+3)(x-3)]/(3x-6)#

If there are any common factors in the numerator and the denominator, we can cancel them.

But, we do not have any.

Hence, we will move on.

Next, we set the denominator to zero.

#color(blue)((3x-6) = 0#

Add #6# to both sides.

#(3x-6+6) = 0+6#

#(3x-cancel 6+cancel 6) = 0+6#

#rArr 3x = 6#

#rArr x = 6/3 = 2#

Hence, our Vertical Asymptote is at #color(blue)(x=2)#

Refer to the graph below:

#color(green)(Step.2):#

To find the Horizontal Asymptote:

Consider:

#color(red)(y = f(x) = (x^2-9)/(3x-6))#

Since the highest degree of the numerator is greater than the highest degree of the denominator,

Horizontal Asymptote DOES NOT EXIST

#color(green)(Step.3):#

To find the Slant/Oblique Asymptote:

Consider:

#color(red)(y = f(x) = (x^2-9)/(3x-6))#

Since, the highest degree of the numerator is one more than the highest degree of the denominator, we do have a Slant/Oblique Asymptote

We will now perform the Polynomial Long Division using
#color(red)(y = f(x) = (x^2-9)/(3x-6))#

Hence, the Result of our Long Polynomial Division is

#x/3 + 2/3 + (-5/(3x-6))#

Equation of the Slant/Oblique Asymptote is

#color(blue)(y = x/3 + 2/3)#

Refer to the graph below:

We have all the required results now.

Benjamin Achtenberg · Answer

undefined To find the vertical asymptote of the function $ \frac{x^2 - 9}{3x-6} $, first, set the denominator equal to zero and solve for $x$. The vertical asymptote occurs where the denominator equals zero. In this case, $3x - 6 = 0$ implies $x = 2$. So, the vertical asymptote is $x = 2$.

To find the horizontal asymptote, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $y = 0$. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. In this case, the degrees of the numerator and denominator are both 1, so the horizontal asymptote is $y = \frac{1}{3}$.

To find oblique asymptotes, perform long division or polynomial division on the function. The quotient will represent the oblique asymptote.