Todd’s Evil Limit – Solution

We start by replacing e by its Taylor Series:

Now a simple, but non-rigorous way to go is to observe that when x is very small, sin(x) is approximately equal to x. If you allow me to use that approximation we obtain:

If that makes you uncomfortable, we can use the more honest, more cumbersome method of L’Hopital’s rule:

And there we are! Three notes:

  1. Todd found this as an exercise in “Excursions in Calculus” by Robert M. Young. He recommends that all of you read this book.
  2. I’ve known Todd longer than you have. He likes me.
  3. Todd is not single. Sorry.