The standard definitions of limit, continuity and derivative are things of beauty mathematically - flexible and well-honed like fine woodworking tools. But to get calculus students to care, and appreciate their meaning and significance, takes some motivation.

A pretty good way to motivate $\epsilon$-$\delta$ is that it has to do with determining what control on input error ($\delta$) is needed to guarantee meeting a given tolerance for output error ($\epsilon$). How accurately do you have to aim a spacecraft to ensure it enters Martian orbit without burning up the way Beagle 2 did, costing hundreds of millions? Students can appreciate this is a serious question, and that it is fair to insist they be able to handle simple examples like $f(x)=-100x+50$, $\epsilon=10^{-2}$. (In large lectures for freshmen, I wouldn't do much more than Lipschitz examples or something carefully designed so $\delta$ is easy to find without cases. Many calculus students are adults but, ahem, need practice with inequalities.)

One can tell engineering students who just want the formulas that they'll be surprised to find that in a couple of years they'll be estimating ``sensitivity coefficients'' numerically from black-box software or experiment. Gee, sensitivity coefficients are just derivatives, and they'll be estimated from the definition, not symbol-pushing.

Speaking of which, it's nice to express the error in the definition explicitly: $$ \frac{f(x)-f(a)}{x-a} = f'(a)+E_a(x), $$ and do the algebra that occurs to few to write $$ f(x)=f(a)+ f'(a)(x-a)+ E_a(x)(x-a) $$ This makes the nature of linear approximation a bit more apparent.