Cogito
Calculus · Chapter 2 · Lesson 2
Types of Discontinuity
Three ways a graph can break.
12 problems · about 21 minutes · LIM-2.A, LIM-2.B
What this lesson teaches
The student classifies discontinuities as removable, jump, or infinite.
- A removable discontinuity is a single hole and can be repaired.
- A jump has two finite one-sided limits that disagree.
- An infinite discontinuity occurs at a vertical asymptote.
Warm Up
Straightforward practice. Get the method working first.
5 problemsLeft limit 3, right limit 3, function value 8. Which kind? 1, 2 or 3.
Answer 1
Why Removable.
Why is a removable discontinuity called removable?
Answer Redefining the function at that one point closes the hole.
Why One redefinition repairs it.
Left limit 4, right limit 4, function undefined. Which kind? 1, 2 or 3.
Answer 1
Why They agree and are finite.
Left limit 1, right limit 6. Which kind? 1, 2 or 3.
Answer 2
Why Finite but unequal.
Values run away near the point. Which kind? 1, 2 or 3.
Answer 3
Why Unbounded.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = (x² − 25) ÷ (x − 5). What repairs it at x = 5?
Answer 10
Why x + 5 at x = 5.
f(x) = 1 ÷ (x − 4). Which kind at x = 4? 1, 2 or 3.
Answer 3
Why A vertical asymptote.
How many kinds of discontinuity are named here?
Answer 3
Why Count them.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each description with its kind of discontinuity.
Answer A single hole in the graph → Removable; The graph steps to a new height → Jump; The graph runs off the screen → Infinite
Why Only one of the three can be repaired.
Which kind can be repaired by redefining one point? 1, 2 or 3.
Answer 1
Why Its name gives it away.
The Hole: f(x) = (x² − 9) ÷ (x − 3). What value at x = 3 would repair the break?
Answer 6
Why 6.
The Step: Left limit 2, right limit 5. Which kind of discontinuity? Enter 1 removable, 2 jump, 3 infinite.
Answer 2
Why A jump.