---
title: Some epsilon-delta proofs
---
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 $\lim_{x\to-4} |2x+5| = 3$.
If $|x+4|<1$, then $-1 -2x - 5 > 3 - \epsilon \\
3 + \epsilon > |2x +5| > 3-\epsilon\\
\epsilon > |2x+5|-3 > \epsilon \\
\big| |2x+5| -3\big| < \epsilon \ \ \ \ \ \square
\end{align}$$