Cogito
Precalculus · Chapter 8 · Lesson 1
The Idea of a Limit
Where a function is heading, not where it lands.
12 problems · about 21 minutes · TEKS P.1.A
What this lesson teaches
The student estimates limits from tables and graphs and distinguishes a limit from a function value.
- A limit describes where a function is heading, not where it lands.
- A limit can exist where the function is undefined.
- The two-sided limit exists only when both one-sided limits agree.
Warm Up
Straightforward practice. Get the method working first.
5 problemsLimit of 4x − 2 as x approaches 3?
Answer 10
Why 10.
A function is undefined at x = 2. Can its limit at 2 exist?
Answer Yes, a limit describes approach, not arrival.
Why Yes. The limit ignores what happens at the point itself.
Limit of 3x as x approaches 4?
Answer 12
Why Substitute.
Limit of x² as x approaches 5?
Answer 25
Why Substitute.
Limit of (x² − 9) ÷ (x − 3) as x approaches 3?
Answer 6
Why Factor to x + 3.
Build It Up
The same ideas with more to keep track of.
3 problemsLimit of 7 as x approaches 2?
Answer 7
Why A constant never changes.
Left limit is 3 and right limit is 5. Does the two-sided limit exist? 1 for yes, 0 for no.
Answer 0
Why They must agree.
Limit of (x² − 25) ÷ (x − 5) as x approaches 5?
Answer 10
Why Factor to x + 5.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich situations mean a limit does not exist?
Answer The two sides approach different values; The values run off to infinity
Why A hole does not stop a function from approaching something.
Limit of x + 8 as x approaches −3?
Answer 5
Why Substitute.
The Approach: What is the limit of 2x + 1 as x approaches 3?
Answer 7
Why 7.
The Hole: What is the limit of (x² − 4) ÷ (x − 2) as x approaches 2?
Answer 4
Why 4.