Problem: Investigate convergence of the following sequence of functions:
Solution: First we investigate pointwise convergence. We treat x as a parameter and evaluate limit with respect to k. Since the sine is bounded by 1 and the factor in front of it goes to zero, we get
Formally one could use for instance comparison.
Conclusion: The given sequence converges to the function
How about uniform convergence? We start by investigating the difference between f and a particular fk on the above region of convergence.
We have just proved uniform convergence.
Conclusion: The given sequence converges to the function ex/4 uniformly on the whole real line.
I asked the computer for a few graphs just to show the general idea of what is happening here.