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Find This Limit Please

  1. Jan 23, 2016 #1
    What is the ##\lim_{x \to \infty} x^2##?

    What I get is:

    ##\lim_{x \to \infty} x^2##
    ##= \lim_{x \to \infty} \frac{\frac{1}{x^2}}{\frac{1}{x^2}} x^2##
    ##= \lim_{x \to \infty} \frac{\frac{x^2}{x^2}}{\frac{1}{x^2}}##
    ##= \lim_{x \to \infty} \frac{1}{\frac{1}{x^2}}##
    ##= \frac{1}{\frac{1}{\infty}}##
    ##= \frac{1}{0}##
     
  2. jcsd
  3. Jan 23, 2016 #2

    PeterDonis

    User Avatar
    2016 Award

    Staff: Mentor

    It looks like you have shown (in a somewhat roundabout way) that the limit is undefined--in other words, that ##x^2## increases without bound as ##x## increases without bound (or, in the more usual sloppy terminology, ##x^2 \rightarrow \infty## as ##x \rightarrow \infty##). Does that seem reasonable to you?
     
  4. Jan 25, 2016 #3

    Mark44

    Staff: Mentor

    Going from this step to the next, you are saying that ##\lim_{x \to \infty} x^2 = \infty##. In other words you have used a number of unnecessary steps to arrive at pretty much the same thing as you started with.

    Also, you should never write either ##\frac 1 {\infty}## or ##\frac 1 0##. In the first, ##\infty## is not a number that can be used in arithmetic expressions, and in the second, ##\frac 1 0## is undefined.
    Much more simply, if x grows large without bound, ##x^2## does the same even more rapidly.
     
  5. Jan 25, 2016 #4

    fresh_42

    Staff: Mentor

    The common ε-δ-definition of limits turns in the case of unlimited sequences to
    ##∀ r ∈ ℝ ∃ N ∈ ℕ ∀ n ≥ N : x_n > r## for ##\lim_{n→∞} x_n = ∞## (##x_n < r## for ##\lim_{n→∞} x_n = -∞##)
     
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