Cogito
AP Calculus AB · Chapter 1 · Lesson 1
What a Limit Says
Where the function is heading, not where it lands.
11 problems · about 24 minutes · AP Calculus AB 1.2, 1.3
What this lesson teaches
The student evaluates limits and distinguishes the limit at a point from the value at that point.
- A limit describes what the outputs approach, not what happens at the point.
- A limit exists only when both one-sided limits agree.
- 0/0 means more work is needed, not that the answer is zero.
Warm Up
Straightforward practice. Get the method working first.
4 problemsLimit of (x² − 16)/(x − 4) as x approaches 4?
Answer 8
Why 8.
Limit of 3x + 1 as x approaches 2?
Answer 7
Why Continuous — just substitute.
Limit of (x² − 1)/(x − 1) as x approaches 1?
Answer 2
Why Factor and cancel.
Left limit is 4, right limit is 4. Does the limit exist? 1 for yes, 0 for no.
Answer 1
Why They agree.
Build It Up
The same ideas with more to keep track of.
3 problemsLeft limit is 2, right limit is 5. Does the limit exist? 1 for yes, 0 for no.
Answer 0
Why They disagree.
Select every statement that is true about limits.
Answer A limit can exist where the function is undefined.; Both one-sided limits must agree.
Why 0/0 is an invitation to do more work, not an answer.
Limit of 5 as x approaches 100?
Answer 5
Why A constant never changes.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsLimit of (x² − 25)/(x − 5) as x approaches 5?
Answer 10
Why x + 5.
Order these steps for a limit that gives 0/0 on substitution.
Answer 1. Substitute and get 0/0 2. Factor the numerator and denominator 3. Cancel the shared factor 4. Substitute again
Why Substitution happens twice.
The Removable Discontinuity: What is the limit of (x² − 9)/(x − 3) as x approaches 3?
Answer 6
Why x + 3 approaches 6.
Value Against Limit: f(2) = 10 but the outputs near 2 approach 3. What is the limit as x approaches 2?
Answer 3
Why 3.