Question

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?

Answer 1

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

Answer 2

We suspect that lim_x→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.