Computing Limits and Indeterminate Forms

What to do when substitution fails.

12 problems · about 22 minutes · LIM-1.D, LIM-1.E

What this lesson teaches

The student evaluates limits algebraically and applies the Squeeze Theorem.

Substitute first; a number is the answer.
0 ÷ 0 means factor, rationalise, or clear the fractions.
The Squeeze Theorem forces a limit when a function is trapped between two others.

Straightforward practice. Get the method working first.

5 problems

Limit of (x² − 1) ÷ (x − 1) as x approaches 1?
Answer 2
Why 2.
What does the Squeeze Theorem require?
Answer The function is trapped between two others sharing a limit.
Why Bounding above and below by functions with a common limit.
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 (x³ − 8) ÷ (x − 2) as x approaches 2?
Answer 12
Why The factor is x² + 2x + 4.

The same ideas with more to keep track of.

3 problems

Limit of (1 − cos x) ÷ x as x approaches 0?
Answer 0
Why The second special limit.
Limit of 5x + 2 as x approaches 3?
Answer 17
Why Substitute.
g is trapped between −x² and x² near 0. What is its limit there?
Answer 0
Why Both bounds go to zero.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each form by what it tells you.
Answer More work needed: Zero over zero, Infinity minus infinity · Limit does not exist: Five over zero, Negative three over zero
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 Special Limit: Limit of sin x ÷ x as x approaches 0?
Answer 1
Why 1.