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Infinite Limits and Vertical Asymptotes

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Finding Limits at Infinity

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Key Questions

  • Answer:

    An infinite limit is what a functions y value approaches as it approaches infinity or negative infinity

    Explanation:

    An infinite limit is what a functions y value approaches as the x value approaches infinity or negative infinity

    For example
    #limxtooo e^x=oo#
    #limxto-oo e^x=0#

  • The vertical asymptote is a place where the function is undefined and the limit of the function does not exist.

    This is because as #1# approaches the asymptote, even small shifts in the #x#-value lead to arbitrarily large fluctuations in the value of the function.


    On the graph of a function #f(x)#, a vertical asymptote occurs at a point #P=(x_0,y_0)# if the limit of the function approaches #oo# or #-oo# as #x->x_0#.

    For a more rigorous definition, James Stewart's Calculus, #6^(th)# edition, gives us the following:

    "Definition: The line x=a is called a vertical asymptote of the curve #y=f(x)# if at least one of the following statements is true:

    #lim_(x->a)f(x) = oo#
    #lim_(x->a)f(x) = -oo#
    #lim_(x->a^+)f(x) = oo#
    #lim_(x->a^+)f(x) = -oo#
    #lim_(x->a^-)f(x) = oo#
    #lim_(x->a^-)f(x) = -oo#"

    In the above definition, the superscript + denotes the right-hand limit of #f(x)# as #x->a#, and the superscript denotes the left-hand limit.

    Regarding other aspects of calculus, in general, one cannot differentiate a function at its vertical asymptote (even if the function may be differentiable over a smaller domain), nor can one integrate at this vertical asymptote, because the function is not continuous there.

    As an example, consider the function #f(x) = 1/x#.

    As we approach #x=0# from the left or the right, #f(x)# becomes arbitrarily negative or arbitrarily positive respectively.

    In this case, two of our statements from the definition are true: specifically, the third and the sixth. Therefore, we say that:

    #f(x) = 1/x# has a vertical asymptote at #x=0#.

    See image below.

    enter image source here

    Sources:
    Stewart, James. Calculus. #6^(th)# ed. Belmont: Thomson Higher Education, 2008. Print.

  • The vertical asymptote of #y=1/(x+3)# will occur when the denominator is equal to 0. In this case, that will occur at -3, so the vertical asymptote occurs at #x=-3#. There is no y-coordinate to be included.

    For a more thorough explanation behind vertical asymptotes, see here: http://socratic.org/questions/what-is-a-vertical-asymptote-in-calculus? In summary however, vertical asymptotes occur at #x#-values where the limit of the function, either overall or from the right or the left, approaches #+-oo#.

Questions

  • George C. answered · 4 months ago