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Sequences and limits

  1. Jan 31, 2010 #1
    1. The problem statement, all variables and given/known data

    [tex] x_{n}(t) \left\{\begin{array}{cc}nt,&\mbox{ if }
    0\leq t \leq \frac{1}{n}\\ \frac{1}{nt} & \mbox{ if } \frac{1}{n}\leq t \leq 1 \end{array}\right. [/tex]

    2. Relevant equations

    3. The attempt at a solution

    Can someone help me get started finding the limit as n -> inf? I've never taken the limit of a sequence that has such a dependence on t.

    For t in [0, (1/n)], the values of the sequence will range between 0 and 1, and for t in [(1/n),1], the values will range between 0 and 1 as well. It doesn't really matter how large you take n...
  2. jcsd
  3. Jan 31, 2010 #2


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    Pick a fixed x0 in [0,1] and think about limit x_n(x0) as n->infinity. If x0 is not zero there is always an N>0 such that 1/N<x0. That means for all n>N the definition of x_n(x0) is 1/(n*x0). What's the limit at x0?
  4. Feb 1, 2010 #3
    What do you mean by pick and x0? You mean, pick a t0?
  5. Feb 1, 2010 #4


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    t0, x0 whatever. Sure, call the point t0 if you want.
  6. Feb 1, 2010 #5
    How about Alfred? Anyway, I think I got what you are saying. No matter what your choice for t, this function will merge to 0 as n -> inf.

    thank you for your time.
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