Cogito
AP Calculus AB · Chapter 1 · Lesson 2
Continuity
Three conditions, all required.
11 problems · about 22 minutes · AP Calculus AB 1.11, 1.13
What this lesson teaches
The student tests continuity at a point and classifies discontinuities.
- Continuity at a point needs the value, the limit, and their agreement.
- A hole is removable; a jump and an asymptote are not.
- Polynomials are continuous everywhere.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(4) = 1 and the limit at 4 is 6. Continuous? 1 for yes, 0 for no.
Answer 0
Why Discontinuous.
f(3) = 4 and the limit at 3 is 4. Continuous? 1 for yes, 0 for no.
Answer 1
Why All three hold.
f(3) undefined, limit is 4. Continuous? 1 for yes, 0 for no.
Answer 0
Why Condition one fails.
Sort each discontinuity by whether redefining one point repairs it.
Answer Removable: A hole · Not removable: A jump, A vertical asymptote
Why Only one of the three has a limit at all.
Build It Up
The same ideas with more to keep track of.
3 problemsIs a polynomial continuous everywhere? 1 for yes, 0 for no.
Answer 1
Why No denominators, no gaps.
1/(x − 6). At which x is it discontinuous?
Answer 6
Why Zero denominator.
A hole at x = 2 with limit 9. What value at 2 would repair it?
Answer 9
Why Match the limit.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsLeft limit 5, right limit 5, f(1) = 5. Continuous at 1? 1 for yes, 0 for no.
Answer 1
Why All three agree.
How many conditions fail at a vertical asymptote where f is undefined?
Answer 3
Why No value, no limit, so no equality either.
The Three Conditions: How many conditions must hold for continuity at a point?
Answer 3
Why Three.
The Jump: Left limit 2, right limit 7. Is this discontinuity removable? 1 for yes, 0 for no.
Answer 0
Why Not removable — it is a jump.