## How to prove a limit exists using the epsilon delta definition of a limit [closed]

I understand how to find a limit. I understand the concept of the epsilon delta definition of a limit. Can you walk me through what we're doing in this worked example? It is from my student solutions manual to my textbook. I need help understanding what we're saying here, and why. I understand the math expressions, but I do not understand why we chose the ones we did, and why and how they prove anything. Can you help?

Find the Lim as x --> 1 of (x+4) and prove it exists using the e-d def of limit where e = epsilon not the famous constant, and d = delta... By direct substitution, lim is 5. Understood. Now, here's where I start to get confused...

Let e > 0 be given.

Choose d = e.

0 < | x-1 | < d = e

| (x+4) - 5 | < e

| f(x) - L | < e

Proved.

Uh, okay, if you say so... Now, what's going on here line by line and term by term?

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If that's all the solution manual says, I'd say that's not a very good solution manual. At any rate, the question is not appropriate for this particular site. I think you'll get good answers if you try at math.stackexchange.com – Tony Huynh Sep 19 2011 at 1:19
Free, you may want to take a glance at the FAQ for this site. Then you will probably find that math.stackexchange is a better place for a question like this. However as someone who teaches mathematics, thank you for caring enough to ask. – BSteinhurst Sep 19 2011 at 1:20
@Tony, we were typing at the same time so I certainly didn't mean to duplicate. – BSteinhurst Sep 19 2011 at 1:21
Sorry guys, I understand what you have said, and thank you for the pointer to the right place. I'd like to delete my Q. – Free Sep 19 2011 at 1:52
@euklid345: but they might not. I don't think that is important in this instance. – Yemon Choi Sep 19 2011 at 2:15
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## 2 Answers

Someone with editing powers, please delete my question. I appreciate having been directed to an appropriate forum. Thank you, folks.

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We suspect that limx→1 (x + 4) = 5, but how do we prove it?  What does it mean for the limit to be 5?  The definition is of the form "for every ε > 0 there exists some δ > 0 such that ... ."  To prove this, they choose an arbitrary ε > 0 and look for a δ > 0 that has the desired property.

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