How do you find the Vertical, Horizontal, and Oblique Asymptote given #f(x)=(2x^3+3x^2+x)/(6x^2+x-1)#?

Question

Anna Allen · Accepted Answer

A vertical assymptote at #x = 1/3# and
a slant assymptote given by #y = x/3+4/9#

#f(x) = (2 x^3 + 3 x^2 + x)/(6 x^2 + x - 1) = 1/3(x (x+1/2)(x+1))/((x-1/3)(x+1/2)) = 1/3(x(x+1))/((x-1/3))#

We have a vertical assymptote at #x = 1/3# and also we know

#1/3x(x+1) = (x-1/3)(ax+b)+c#

Equating power coefficients

#{ (b/3 - c=0), ( 1/3 + a/3 - b=0), (1/3 - a=0) :}#

and solving

#a = 1/3,b=4/9,c=4/27#

The assymptote is given by

#y = x/3+4/9#

Attached a figure with vertical and slant assymptotes

Ariana Alvarez · Answer

undefined To find the vertical, horizontal, and oblique asymptotes of the function $ f(x) = \frac{2x^3 + 3x^2 + x}{6x^2 + x - 1} $:

1. **Vertical Asymptotes**: Vertical asymptotes occur where the denominator of the function becomes zero but the numerator does not. Set the denominator equal to zero and solve for $ x $. Any solution found will represent a vertical asymptote.

2. **Horizontal Asymptotes**: Horizontal asymptotes occur when $ x $ approaches positive or negative infinity. To find horizontal asymptotes, compare the degrees of the numerator and denominator polynomials. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

3. **Oblique Asymptotes**: Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find the oblique asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. The quotient obtained represents the oblique asymptote.

Let's apply these steps to the given function $ f(x) = \frac{2x^3 + 3x^2 + x}{6x^2 + x - 1} $:

1. **Vertical Asymptotes**:
   Set the denominator $ 6x^2 + x - 1 $ equal to zero and solve for $ x $ to find vertical asymptotes, if any.

2. **Horizontal Asymptotes**:
   Compare the degrees of the numerator and denominator polynomials.

3. **Oblique Asymptotes**:
   If the degree of the numerator is one greater than the degree of the denominator, perform polynomial long division or synthetic division to find the oblique asymptote.