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Homework Help: Show a limit = infinity

  1. Sep 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Show lim [tex]n^{1/2}[/tex]=[tex]\infty[/tex] using the definition of lim [tex]x_{n}[/tex]=[tex]\infty[/tex].



    2. Relevant equations



    3. The attempt at a solution
    We want to show [tex]\left|x^{1/2}-\infty\right|[/tex],[tex]\epsilon[/tex]. I get this far and i hit a blank.
     
  2. jcsd
  3. Sep 28, 2010 #2

    Mark44

    Staff: Mentor

    No, you can't use this [tex]|x^{1/2}-\infty|[/tex]. The definition for the limit of an unbounded function doesn't use infinity or epsilon. Do you know what that definition is?
     
  4. Sep 28, 2010 #3
    Well, I was supposed to supply the definition of lim [tex]x_{n}[/tex]=[tex]\infty[/tex].

    I guess I wasn't so sure on how to do that.
     
  5. Sep 28, 2010 #4

    Mark44

    Staff: Mentor

    No, you're supposed to use the definition of [tex]\lim_{n \to \infty} x_n = \infty[/tex]. That's different.

    [tex]\lim_{x \to \infty} f(x) = \infty[/tex] is defined this way:
    [tex]\forall M \exists N > 0 \ni \forall x > N, f(x) > M[/tex]

    In other words, no matter how large an M someone chooses, there is some number N so that if x > N, then f(x) > M. In other, other words, if you want to get a larger value for f(x), take a larger value for x.

    The definition is similar for your sequence.
     
  6. Sep 28, 2010 #5
    Sorry, I was just using what my book stated and the book didn't include that extra part.
     
  7. Sep 29, 2010 #6

    Mark44

    Staff: Mentor

    Do you still have a question? I can't tell.
     
  8. Sep 29, 2010 #7
    I guess my question is how to use the definition to prove the limit of my original sequence?
     
  9. Sep 29, 2010 #8

    Mark44

    Staff: Mentor

    You work backward. Assuming for the moment that n1/2 > M, can you find some other number N so that when n > N, then n1/2 > M? That's basically what I said in post #4.
     
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