ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

1.10 Exploring Types of Discontinuities

#discontinuities

โฑ๏ธย ย 2 min read

written by

Anusha Tekumulla

anusha tekumulla


Discontinuities

๐ŸŽฅWatch: 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.ย 

There are four types of discontinuities you have to know: jump, point, essential, and removable.ย 

Jump Discontinuity โ†”

If the left and right-handed limits arenโ€™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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fjump-discontinuity.png?alt=media&token=2d20b0c6-184d-4ed2-b9b0-e3fdf8e52a23

Math Warehouse

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.

Point Discontinuity ๐Ÿ•ณ

A point discontinuity occurs when the function has a โ€œholeโ€ in it from a mission point because the function has a value at that point that is โ€œoff the curve.โ€

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fdownload-1.png?alt=media&token=27457277-a436-4a77-be6e-55a3f5697a15

Wikipedia

Essential Discontinuity โ™พ๏ธ

An essential discontinuity occurs when the curve has a vertical asymptote. This is also called an infinite discontinuity.ย 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fdownload-2.png?alt=media&token=d74b5fae-65d0-4539-9a1e-98a19c24db70

Math Warehouse

Removable Discontinuity ๐Ÿšซ

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 โ€œremovable.โ€ย 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fdownload-3.png?alt=media&token=d3a817be-fb5d-40c4-aed0-a6e48c2f68d3

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