How do you find the asymptotes of a rational function?
1 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
*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
*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:
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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^2x12)# (also#y=(x+1)/((x+3)(x4))# ) 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)/(x2)# 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.