Using the Definition of Continuity in Calculus
In this video we do four problems trying to show that a piecewise function is continuous or not by using the definition of continuity at a point (limit at the point equals the value of the function at the point). All of the problems involve using multiple representations of a function (a piecewise defined function notation and a graph of the function parts).
These problems were pulled from a document that CB released during 2020.
Some time stamps if you're looking to jump around:
[0:29] Problem a, showing the function is continuous
[3:18] Problem b, showing a function is not continuous
[5:44] Problem c, showing a function (the reciprocal of the last function) is continuous
[8:12] Problem d, the hard question! We use L'Hospital's Rule to evaluate the limit (which takes a lot of work) and then find c to make the function continuous by thinking about the definition of continuity at a point.
All of these problems are the kind of thing you would see on multiple choice or free response (FRQ) questions on the AP Calculus AB or BC exams. The topics kind of come from Unit 1 (1.11) but for part d you'll need L'Hospital's rule, which is Unit 4 (4.7).