I would like to continue the discussion started in this post by offering you a set of even more challenging limits. (As an additional exercise try solving these limits without Riemann Sums) Lets proceed to our first not so trivial example.
Problem 4. Find the following limit
Solution. We can't apply our tactic directly because we have to deal with product rather than sum. However, as many of you can guess, there is a natural way to transform the product into sum.
Now we see a familiar construction, which can be easily evaluated, namely
Now returning to the initial problem we get
Almost the same problem was proposed at National Mathematics Olympiad in Moldova in 2007.
Problem 5. (Moldova, 2007, class) Calculate the limit
I also would like to propose another problem from Moldova, this time it's District Mathematics Olympiad, 2013. It's much easier than examples discussed in this post, but still serves a decent technique trainer.
Problem 6. (Moldova, 2013, class) Find the limit