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