What are the most interesting limit problems you've run into? I am helping a friend of mine run facilitated study group for 1st year calculus students. In class right now they are covering how to evaluate different types of limits. The examples in the textbook they are using only has easy limit questions. I was wondering what are some interesting difficult limit questions that you have run into where the answer isn't immediately clear? I want to show some of my students examples of harder problems and challenge them instead of asking them trivial problems.   
 A: (1) Construct a sequence that has 2 cluster points.
(2) If a sequence $x_{n}$ converges to $x$, prove that the sequence of the $n$-th averages,
$$
y_{n} = {1 \over n} \sum_{k = 1}^{n} x_{k}
$$
also converges to $x$.
(3) Background: Tossing a coin infinitely many times, we get a sequence of T's and H's.  If we now construct the sequence of the "relative frequences" of T, i.e. the sequence of the ratios
$$
\phi_{n} = {\mbox{number of T's among the first $n$ terms of the sequence} \over n},
$$
then we should have $\lim_{n \rightarrow \infty} \phi_{n} = $ (the probability of getting a T).  This is a special case of the Law of Large Numbers.
Problem: construct a sequence of T's and H's in which the sequence of the $\phi_{n}$'s fails to converge.  Understand why such a sequence could not have resulted from tossing a coin infinitely many times.
(4) Construct a sequence of functions on $[0, 1]$ that converges pointwise, but not uniformly.  (Not usually taught in first semester calculus, but it is within those students' reach.)
