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