Cogito
Precalculus · Chapter 8 · Lesson 2
Computing Limits
Factor, cancel, and then substitute.
12 problems · about 22 minutes · TEKS P.1.A
What this lesson teaches
The student evaluates limits algebraically using factoring, and evaluates limits at infinity for rational functions.
- For continuous functions, substitute and you are done.
- 0 ÷ 0 is indeterminate: factor, cancel, then substitute.
- At infinity, only the highest-degree terms of a rational function matter.
Warm Up
Straightforward practice. Get the method working first.
5 problemsLimit of (x² − 4) ÷ (x − 2) as x approaches 2?
Answer 4
Why 4.
What does the form 0 ÷ 0 tell you?
Answer Nothing yet; more algebra is needed.
Why It is indeterminate, so the work is not finished.
Limit of (x² − 1) ÷ (x − 1) as x approaches 1?
Answer 2
Why Factor to x + 1.
Limit of (x² − 36) ÷ (x − 6) as x approaches 6?
Answer 12
Why Factor to x + 6.
Limit of (6x² + x) ÷ (3x² − 2) as x grows without bound?
Answer 2
Why 6 ÷ 3.
Build It Up
The same ideas with more to keep track of.
3 problemsLimit of (x + 1) ÷ (x² + 3) as x grows without bound?
Answer 0
Why The bottom has the larger degree.
Limit of (10x² + 5) ÷ (4x² + 1) as x grows without bound?
Answer 2.5
Why 10 ÷ 4.
Is 0 ÷ 0 an answer or a signal to do more work? Enter 1 for answer, 2 for signal.
Answer 2
Why It is called indeterminate for a reason.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each degree comparison by the limit at infinity it produces.
Answer Limit is 0: Bottom degree larger, Degree 1 over degree 3 · Limit is a ratio of coefficients: Degrees equal, Degree 2 over degree 2
Why A bigger bottom always wins the race.
Limit of (x² − 49) ÷ (x − 7) as x approaches 7?
Answer 14
Why Factor to x + 7.
The Cancel: Limit of (x² − 16) ÷ (x − 4) as x approaches 4?
Answer 8
Why 8.
The Far End: Limit of (4x² + 1) ÷ (2x² + 3) as x grows without bound?
Answer 2
Why 2.