Basic $\epsilon$-$\delta$ thinking is easy to motivate by using the concepts of input control and output error tolerance. If $f(a)=c$, how accurately do you need to control the input (by specifying $\delta$ and requiring $|x-a|<\delta$) to guarantee you meet a given tolerance for output error ($|f(x)-c|<\epsilon$)?
Week 2 of freshman calculus is not too soon to insist US students learn to answer questions about this topic for simple examples. It's clearly a relevant skill for eventually addressing questions like How accurately do you need to aim a spacecraft to safely enter orbit around Mars?'' (Recall, failing to do that once cost NASA around a half-billion dollars.) Or, for designing safety margins in engineering as Vectornaut suggests.
The significance and usefulness of fundamental calculus concepts is often underappreciated not only by beginning students. A graduate engineering student once explained to me his recent realization that the sensitivity coefficients'' used everywhere in his engineering courses were nothing but derivatives!