Continuity

Three conditions, all required.

10 problems · about 20 minutes · LIM-2.A, LIM-2.B

NameDate

Figure — use these to answer the problems

The two branches never join. That gap is an infinite discontinuity.

y = 1/x + 0

Numerator1

Shift up by0

Figure 1

From earlier lessons. Loosen up before the new work.

2 problems

From “Limits at Infinity and Asymptotes”
Limit of (8x + 3) ÷ (4x − 1) as x grows without bound?
Answer
From “Computing Limits Algebraically”
Limit of (x² − 1) ÷ (x − 1) as x approaches 1?
Answer

Straightforward practice. Get the method working first.

4 problems

f(x) = x³ − 4x. What is f(3)?
Answer
4
What does the Intermediate Value Theorem guarantee?
- A continuous function passes through every value between two of its values.
- Every function has a root.
f(x) = x² − 3. What is f(2)?
Answer
f(x) = x² − 3. What is f(1)?
Answer

The same ideas with more to keep track of.

2 problems

y = 1/(x − 4). At what x is there an infinite discontinuity?
Answer
8
Can a function have a limit at a point where it is undefined?
- Yes. That is a removable hole.
- No.

Mixed problems. Work out what kind of question it is before you start.

2 problems