I was curious to see if I could manage differentiating the hyperbolic sine using the difference quotient. The algebra has been easy, and I've broken it down to the limiting coefficients. Problem is... I realize I don't know how to do epsilon-delta limits! (except for very trivial ones) Before we get there, here's my derivative...

Given

Skipping the few intermediate steps...

I know I'm shooting for the hyperbolic cosine , so now it's just a matter of showing that those limits are both 1. And the only way to do this right is through epsilon-delta proofs.

When I started learning single-variable calculus ahead of time, I sort of skipped over doing the actual proofs and instead just assumed their expected values for the expected result. Though I understood the concept behind it, I pretty much went on to learn differentiation and integration without a clue as to how to do these proofs (why? I don't know!). Now I've stumbled back upon this gap, and I'm hoping that a walkthrough with this particular problem will help me to understand them. Let's start with the first coefficient...

Show that

If

then

I know I have to relate epsilon and delta somehow, but I'm not sure where to even start. Hints/suggestions?