• Support PF! Buy your school textbooks, materials and every day products Here!

Sequences and limits

  • #1

Homework Statement



[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]


Homework Equations





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...
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
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
What do you mean by pick and x0? You mean, pick a t0?
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
What do you mean by pick and x0? You mean, pick a t0?
t0, x0 whatever. Sure, call the point t0 if you want.
 
  • #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.
 

Related Threads on Sequences and limits

Replies
3
Views
3K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
1
Views
737
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
20
Views
3K
  • Last Post
Replies
11
Views
1K
Top