Day 12: Limits of Piecewise Functions

First I show them the limit as x approaches the middle of one piece of the piecewise. Those are easy because all they have to do is plug and chug. Then of course you must show them a limit problem (maybe using the same piecewise as before) but this the limit approaches the cutoff of the piecewise. They need to see that the left and right aren't always going to be the same value. If you have already taught them "jump discontinuities," great. If you haven't, it's good to use the words now in class as a preview of the vocabulary from the continuity lesson.

Alright now that piecewise is out of the way, it's time for absolute value functions. They may or may not have experience writing them out as piecewise functions, so you may need teach that from scratch unless you already did it in the summer assignment review. Tell them that the cutoff this time is going to come from the horizontal shifts. If they limit isn't approaching the cutoff, the problem is easy! If it's the absolute value of x - 4 and the limit approaches the cutoff four, then, it's possible it's going to be DNE but they should do the limits on both sides of four to make sure.

The textbook we used, by James Stewart, puts the Floor Function in this section, but he calls it the Greatest Integer Function. I do this because some of them have never seen a step function before, some of them might go into programming, and because it's a good example of when the left hand side of the limit doesn't equal the right hand side.

On block schedule, this is just half the day (45 min).