Sequences and limits

1. Somefantastik

231
1. The problem statement, all variables and given/known data

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

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. Dick

25,619
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?

3. Somefantastik

231
What do you mean by pick and x0? You mean, pick a t0?

4. Dick

25,619
t0, x0 whatever. Sure, call the point t0 if you want.

5. Somefantastik

231
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.