# Thread: Epsilon-Delta Limits for a Problem

1. 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?

2. Actually, I might have it... (editing)

UPDATE: I'm simply mirroring the example from #4 >here<.

(I'll use in place of to make the formulae a bit lighter)

Proving the limit

Show that: IF , THEN

-----------------------------------------------------------------------------------------

Multiplying both sides by ...

Now, is derived, and I could relate epsilon and delta. However, the right side is still in terms of , meaning I'll have to simplify the problem. Because the limit deals with being close to , we can make the restriction , so that .

With the original inequality , the denominator of the right side is a globally increasing function. So when is at a maximum the quotient as a whole will be at a minimum.

The maximum of the denominator function is , therefore...

Then we are left with two inequalities, and

Given , let . (so that both inequalities are satisfied)

I assume I now have to work each case and put it into the form , which should complete the proof. Am I on the right track?

 Posting Permissions
 You may not post new threads You may not post replies You may not post attachments You may not edit your posts   BB code is On Smilies are On [IMG] code is On [VIDEO] code is On HTML code is Off Trackbacks are Off Pingbacks are Off Refbacks are On Forum Rules