Cogito
Calculus · Chapter 1 · Lesson 2
Computing Limits Algebraically
When substitution fails, factor.
12 problems · about 22 minutes · LIM-1.D, LIM-1.E
What this lesson teaches
The student evaluates limits using substitution, factoring, and rationalising.
- Substitute first; if it gives a number, that is the limit.
- 0 ÷ 0 means factor, rationalise, or clear the fractions, then substitute.
- A non-zero number over zero means the limit does not exist.
Warm Up
Straightforward practice. Get the method working first.
5 problemsLimit of (x² − 1) ÷ (x − 1) as x approaches 1?
Answer 2
Why 2.
Substituting gives 0 ÷ 0. What does that mean?
Answer The top and bottom share a factor that must be cancelled.
Why A shared factor is hiding the answer.
Limit of (x² − 4) ÷ (x − 2) as x approaches 2?
Answer 4
Why Factor to x + 2.
Limit of (x² − 25) ÷ (x − 5) as x approaches 5?
Answer 10
Why Factor to x + 5.
Limit of 5x − 1 as x approaches 4?
Answer 19
Why Substitute.
Build It Up
The same ideas with more to keep track of.
3 problemsLimit of (x² − x − 6) ÷ (x − 3) as x approaches 3?
Answer 5
Why Factor to x + 2.
Limit of (x³ − 8) ÷ (x − 2) as x approaches 2?
Answer 12
Why The factor is x² + 2x + 4.
Limit of (√(x + 9) − 3) ÷ x as x approaches 0, as a decimal?
Answer 0.1667
Why Rationalise to 1 ÷ (√(x+9) + 3), then substitute.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each form by what it tells you.
Answer More work needed: 0 divided by 0, infinity minus infinity · Limit does not exist: 5 divided by 0, −3 divided by 0
Why Only a zero on top makes the form indeterminate.
Limit of (x² − 16) ÷ (x + 4) as x approaches −4?
Answer -8
Why Factor to x − 4.
The Cancel: Limit of (x² − 9) ÷ (x − 3) as x approaches 3?
Answer 6
Why 6.
The Substitution: Limit of (3x + 5) as x approaches 2?
Answer 11
Why 11.