Cogito
Calculus · Chapter 2 · Lesson 1
Continuity
Three conditions, all required.
10 problems · about 20 minutes · LIM-2.A, LIM-2.B
What this lesson teaches
The student tests continuity at a point and classifies discontinuities.
- Continuity needs a value, a limit, and their agreement.
- Breaks come as holes, jumps, or asymptotes.
- A continuous function cannot skip a value.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Limits at Infinity and Asymptotes: Limit of (8x + 3) ÷ (4x − 1) as x grows without bound?
Answer 2
Why 2.
Review — Computing Limits Algebraically: Limit of (x² − 1) ÷ (x − 1) as x approaches 1?
Answer 2
Why 2.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = x³ − 4x. What is f(3)?
Answer 15
Why 15.
What does the Intermediate Value Theorem guarantee?
Answer A continuous function passes through every value between two of its values.
Why It cannot skip an intermediate value.
f(x) = x² − 3. What is f(2)?
Answer 1
Why 4 − 3.
f(x) = x² − 3. What is f(1)?
Answer -2
Why 1 − 3.
Build It Up
The same ideas with more to keep track of.
2 problemsy = 1/(x − 4). At what x is there an infinite discontinuity?
Answer 4
Why Denominator zero.
Can a function have a limit at a point where it is undefined?
Answer Yes. That is a removable hole.
Why The limit ignores the point itself.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Value: f(x) = x³ − x − 1. What is f(2)?
Answer 5
Why 5.
The Conditions: How many conditions must hold for continuity at a point?
Answer 3
Why 3.