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Sequences and limits

by Somefantastik
Tags: convergence, limits, sequences
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Somefantastik
#1
Jan31-10, 04:16 PM
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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...
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Dick
#2
Jan31-10, 04:29 PM
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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?
Somefantastik
#3
Feb1-10, 11:14 AM
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What do you mean by pick and x0? You mean, pick a t0?

Dick
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Feb1-10, 01:49 PM
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Sequences and limits

Quote Quote by Somefantastik View Post
What do you mean by pick and x0? You mean, pick a t0?
t0, x0 whatever. Sure, call the point t0 if you want.
Somefantastik
#5
Feb1-10, 01:53 PM
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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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