# Some epsilon-delta proofs

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