Calculus traditionally begins with a look at limits and continuity. In many courses this is the only time students will see discontinuous functions with removable discontinuities and most won’t see jump discontinuities elsewhere. So why do students have to learn about limits so early?
We tell them that they can’t do derivatives without limits. That, of course, is a lie. After a day or so of fussing with calculating derivatives using the limit definition we introduce them to shortcuts and algorithms. In fact, if someone came up to us and asked us out of the blue “what’s the derivative of [insert favorite elementary function here]“, we would not reach for the nearest limit.
We could resort to our favorite standby answer and say “because it’s going to be on the exam”. That doesn’t feel very satisfying though.
For me, limits are all about expectations. They are about predicting the way the world should work if everything is right. As x gets close to a certain value, does f(x) seem to hover at some level or does it fluctuate wildly like sin 1/x as x->0? Does it approach the same value from the left and the right or does there seem to be a jump as there is in f'(x) at 0 for f(x) = |x|? Somehow even |x| seems contrived to some students, but what about f(x)=x^(2/3). Near x=0 what happens to it’s derivative?
Limits let us know we aren’t in Kansas any more.
You certainly need limits if you’re going to prove any interesting theorems – but rather than expecting and entertaining questions like “when are we ever going to need this stuff”, let’s work to get our students to enjoy limits for what they are.