How do you find the asymptotes of a rational function?

Question

Elijah Allen · Accepted Answer

To Find Vertical Asymptotes: In order to find the vertical asymptotes of a rational function, you need to have the function in factored form. You also will need to find the zeros of the function. For example, the factored function #y = (x+2)/((x+3)(x-4)) # has zeros at x = - 2, x = - 3 and x = 4. *If the numerator and denominator have no common zeros, then the graph has a vertical asymptote at each zero of the denominator. In the example above #y = (x+2)/((x+3)(x-4)) #, the numerator and denominator do not have common zeros so the graph has vertical asymptotes at x = - 3 and x = 4. *If the numerator and denominator have a common zero, then there is a hole in the graph or a vertical asymptote at that common zero.
Examples: 
1. #y= ((x+2)(x-4))/(x+2)# is the same graph as y = x - 4, except it has a hole at x = - 2.  2.#y= ((x+2)(x-4))/((x+2)(x+2)(x-4))# is the same as the graph of #y = 1/(x + 2),# except it has a hole at x = 4. The vertical asymptote is x = - 2. To Find Horizontal Asymptotes: The graph has a horizontal asymptote at y = 0 if the degree of the denominator is greater than the degree of the numerator. Example: In #y=(x+1)/(x^2-x-12)# (also #y=(x+1)/((x+3)(x-4))# ) the numerator has a degree of 1, denominator has a degree of 2. Since the degree of the denominator is greater, the horizontal asymptote is at #y=0#.   If the degree of the numerator and the denominator are equal, then the graph has a horizontal asymptote at #y = a/b#, where a is the coefficient of the term of highest degree in the numerator and b is the coefficient of the term of highest degree in the denominator. Example: In #y=(3x+3)/(x-2)#  the degree of both numerator and denominator are both 1, a = 3 and b = 1 and therefore the horizontal asymptote is #y=3/1# which is #y = 3# If the degree of the numerator is greater than the degree of the denominator, then the graph has no horizontal asymptote.

Sadie Alexander · Answer

undefined To find the asymptotes of a rational function, follow these steps:

1. **Vertical Asymptotes**: Set the denominator equal to zero and solve for $x$. The vertical asymptotes occur at the values of $x$ that make the denominator zero, unless they are canceled out by factors in the numerator.

2. **Horizontal Asymptotes**:
   - 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 is equal to the degree of the denominator, divide the leading coefficients of both polynomials to find the horizontal asymptote.
   - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

3. **Oblique (Slant) Asymptotes**: If the degree of the numerator is exactly one more than the degree of the denominator, perform polynomial long division to divide the numerator by the denominator. The quotient obtained will be the equation of the oblique asymptote.

4. **Holes**: Factor both the numerator and the denominator, and cancel out any common factors. The values of $x$ that cancel out to create holes in the graph of the function.

By determining these types of asymptotes, you can understand the behavior of the rational function as $x$ approaches infinity or negative infinity, or where it may have discontinuities.