🎥Watch: AP Calculus AB/BC - Continuity, Part II
This is the first topic dealing with continuity in unit 1. Until this point, our main focus was limits and how to determine them. However, a large part in finding and determining limits is knowing whether or not the function is continuous at a certain point. In topics 1.9 - 1.13, we will discuss continuity and different types of discontinuities you will see on the AP Exam.
There are four types of discontinuities you have to know: jump, point, essential, and removable.
If the left and right-handed limits aren’t equal, then the double-handed limit does not exist (DNE). In this scenario, we would have a jump discontinuity. The graph of that would look something like this:
As you can see, there is a break in the physical continuity of the graph at x. If the limit from both sides is different, how can we find a specific value that the function approaches? We cannot. Thus, the limit DNE.
A point discontinuity occurs when the function has a “hole” in it from a mission point because the function has a value at that point that is “off the curve.”
An essential discontinuity occurs when the curve has a vertical asymptote. This is also called an infinite discontinuity.
This last discontinuity is fairly common. A removable discontinuity occurs when you have a rational expression with a common factors in the numerator and denominator. Because these factors can be cancelled, the discontinuity is “removable.”
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1.10Exploring Types of Discontinuities
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