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

Vertical Asymptote:

Horizontal Asymptote: None

Equation of the Slant/Oblique Asymptote:

Given:

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:

We will factor where possible:

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.

Add

Hence, our Vertical Asymptote is at

Refer to the graph below:

To find the Horizontal Asymptote:

Consider:

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

Horizontal Asymptote DOES NOT EXIST

To find the Slant/Oblique Asymptote:

Consider:

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

Hence, the Result of our Long Polynomial Division is

Equation of the Slant/Oblique Asymptote is

Refer to the graph below:

We have all the required results now.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the asymptotes for #f(x)= 1/x^2#?
- How do you find vertical, horizontal and oblique asymptotes for #f(x) = (x^2-2x+6) / (x+6)#?
- How do you find the asymptotes for #g(x)=(3x^2+2x-1)/(x^2-4)#?
- How do you find the Vertical, Horizontal, and Oblique Asymptote given #f(x)=(x^3-x)/(x^3-4x)#?
- How do you identify all asymptotes or holes for #g(x)=(x^3+3x^2-16x+12)/(x-2)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7