Asymptotes

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Finding horizontal and vertical asymptotes
11:22 — by Khan Academy

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Key Questions

  • 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 by#0#)

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

  • 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.

  • 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.

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