Cogito
Calculus · Chapter 1 · Lesson 1
What a Limit Is
Where a function is heading, not where it arrives.
8 problems · about 21 minutes · LIM-1.A, LIM-1.C
What this lesson teaches
The student evaluates limits and distinguishes a limit from a function value.
- A limit is where a function heads, not what it equals.
- A function need not be defined at a point to have a limit there.
- If the two one-sided limits differ, the limit does not exist.
Warm Up
Straightforward practice. Get the method working first.
4 problemsLimit of (x² − 36)/(x − 6) as x approaches 6?
Answer 12
Why 12.
What does a limit describe?
Answer Where the function is heading as x approaches the point.
Why Where it heads, not where it lands.
Limit of 2x − 3 as x approaches 4?
Answer 5
Why Substitute.
Limit of (x² − 9)/(x − 3) as x approaches 3?
Answer 6
Why Factor and cancel.
Build It Up
The same ideas with more to keep track of.
2 problemsLimit of (x² − 25)/(x − 5) as x approaches 5?
Answer 10
Why x + 5.
Must a function be defined at a point to have a limit there?
Answer No.
Why The hole example.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Limit: The limit of (x² − 4)/(x − 2) as x approaches 2. What is it?
Answer 4
Why 4.
The Direct One: The limit of 3x + 1 as x approaches 5. What is it?
Answer 16
Why 16.