# How to go about proving a limit by the epsilon-delta definition? What to do first?

Like the title says, I'm never sure what to do when first presented with a proof like these. I've seen a few examples, and to me, it's as if most of the time professors just pull something out of their bag and start manipulating the delta and epsilon inequalities trying to write one in terms of the other, but there never seems to be any pattern to it.

I know these proofs aren't usually methodical, but what I'm trying to say is that I just don't know how to begin the proof on my own. Any tips to overcome this problem? How do you guys usually tackle these proofs? Thanks a lot!

By the way, I'm doing multivariable calculus, so I'm mostly concerned with multivariable Epsilon-delta proofs (in one variable they're not that bad, though I have little sympathy for them too).

• This question is probably too vague, could you add a specific question so you're more on topic and it doesn't get flagged? Sep 25 '16 at 0:45

$\newcommand{\eps}{\varepsilon}\newcommand{\Vec}{\mathbf{#1}}$To be brutally terse:

• Phase 1: The "scratch work" or "back story":

1. Decide the (probable) value $L = \lim(f, \Vec{x}_{0})$ of the limit by any means.

2. Fix $\eps > 0$. Write down the inequality $|f(\Vec{x}) - L| < \eps$, and work backward, trying to obtain a sufficient condition of the form $\|\Vec{x} - \Vec{x}_{0}\| < \phi(\eps)$ for some positive function $\phi$.

• Phase 2: The final draft.

1. Fix $\eps > 0$ arbitrarily, put $\delta = \phi(\eps) > 0$, and write up the proof that $\|\Vec{x} - \Vec{x}_{0}\| < \delta$ implies $|f(\Vec{x}) - L| < \eps$.

Step 3. establishes that "For every $\eps > 0$, there exists a $\delta > 0$ such that $\|\Vec{x} - \Vec{x}_{0}\| < \delta$ implies $|f(\Vec{x}) - L| < \eps$."

• That's a good strategy, but I actually think I still struggle when following these steps. It's the second part of phase 1 that gives me trouble. How do I work backward? Sometimes it's not too clear what I can do from the inequality $|f(\Vec{x}) - L| < \eps$ Sep 25 '16 at 15:20
• That step is usually where most of the work gets expended, and the precise form of $f$ is crucial. It's difficult to be more specific about strategy without a particular function in mind, as 3-in-441's comment notes. In case it helps, Spivak's Calculus observes that theorems about limits of sums, products, quotients, and compositions amount to general strategies for working backward to choose $\delta$. The proofs of these theorems contain useful idioms, mostly coming from the triangle and reverse triangle inequalities. Sep 25 '16 at 15:37
• Hmm so I should check out Spivak's book for info on solving this step? I know I should've added examples, but I have such a varied set of functions to prove that I wanted something a bit more general. Sep 25 '16 at 16:20