Infinite Limits and Vertical Asymptotes

Key Questions

  • What is a vertical asymptote in calculus?

    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.

    Psykolord1989 . · · Aug 24 2014
  • What is the vertical asymptote of y=1/(x+3) ?

    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.

    Psykolord1989 . · 1 · Aug 24 2014
  • What is an infinite limit?

    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

    Rohan N. · 1 · Mar 29 2018

Questions

Limits