Some epsilon-delta proofs

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Update (May 2022): Like a lot of things in math, the main idea here now seems obvious but it was not at all obvious to me when I was first learning it. I’d probably write the page/proof very differently now, but I’m just leaving this up as a piece of history.

From Salas’s Calculus, 10th edition, page 104: Chapter 2 review exercise 45. Below, the important thing to keep in mind is that we want to use the “piecewise function idea”: that if a function can be thought of as a piecewise function, we first want to restrict it to where it is essentially nonpiecewise, and then show that the limit exists there.

Proof. We want to show . If , then so so , which means is negative. But then . So we want to show that for all satisfying , that holds. But using what we know, is the same thing as as long as . So let . Then so But since , we know So i.e. . From we have then