Asymptotes

Key Questions

  • How do you find the asymptotes of a rational function?

    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.

    Elizabeth P. · 2 · Sep 15 2014
  • What are some examples of functions with asymptotes?

    Example 1: f(x)=x^2/{(x+2)(x-3)}

    Vertical Asymptotes: x=-2 and x=3
    Horizontal Asymptote: y=1
    Slant Asymptote: None

    Example 2: g(x)=e^x

    Vertical Asymptote: None
    Horizontal Asymptote: y=0
    Slant Asymptote: None

    Example 3: h(x)=x+1/x

    Vertical Asymptote: x=0
    Horizontal Asymptote: None
    Slant Asymptote: y=x

    I hope that this was helpful.

    Wataru · · Sep 27 2014
  • What is an asymptote?

    An asymptote is a value of a function that you can get very near to, but you can never reach.

    Let's take the function y=1/x
    graph{1/x [-10, 10, -5, 5]}
    You will see, that the larger we make x the closer y will be to 0
    but it will never be 0 (x->oo)

    In this case we call the line y=0 (the x-axis) an asymptote

    On the other hand, x cannot be 0 (you can't divide by0)

    So the line x=0 (the y-axis) is another asymptote.

    MeneerNask · 2 · Jan 28 2015

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Functions Defined and Notation