Always substitute first. For a continuous function the limit is the value. 0 ÷ 0 reports that a shared factor is hiding the answer: factor and cancel, rationalise with a conjugate, or clear a compound fraction. The Squeeze Theorem handles what algebra cannot.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Factor and cancel
(x² − x − 6)/(x² − 4) as x → 2: (x − 2)(x − 3)/((x − 2)(x + 2)) = (x − 3)/(x + 2) → −1/4. Canceling changes the function only at the one point the limit ignores.
A compound fraction
(1/x − 1/3)/(x − 3) as x → 3: the top is (3 − x)/(3x), so the whole is −(x − 3)/(3x(x − 3)) = −1/(3x) → −1/9.
The Squeeze Theorem
If g is trapped between f and h near a point, and f and h share a limit there, g is forced to that same limit. −x² ≤ x² sin(1/x) ≤ x², and both bounds → 0, so x² sin(1/x) → 0 as x → 0, though it oscillates wildly.
The special limits
sin x/x → 1 and (1 − cos x)/x → 0 as x → 0. Check: sin 0.01/0.01 = 0.99998. They appear in every derivation of the trigonometric derivatives and on the exam every year.
Using them
sin 3x/x as x → 0: write it as 3 · sin 3x/(3x). As x → 0, 3x → 0, so sin 3x/(3x) → 1 and the limit is 3. Match the inside of the sine to the denominator.
Not all zeros are equal
5/0 is not indeterminate: the values run away and the limit does not exist. ∞/∞ and 0 × ∞ are indeterminate; 5/0 and 5/∞ are not. Name the form before choosing a method.
Step 2: Try It Yourself
Tap and try it out.
Find the limit of (x² − x − 6)/(x² − 4) as x → 2, and of sin 3x/x as x → 0
- 1x = 2: (4 − 2 − 6)/(4 − 4) = 0/0indeterminate
Step 0 of 4
- Point(1, 2)
Step 3: In Real Life
A formula at its edge
Average cost at zero items is 0/0. Factor and cancel to see the cost of the first unit. The Squeeze Theorem does the same job for an oscillating signal a sensor cannot resolve.
Step 4: Watch an Example
One step at a time.
Watch Priya Rationalise a Limit
Priya needs the limit of (√(x + 4) − 2) ÷ x as x approaches 0.
- Step 1
Substituting 0 gives (2 − 2) ÷ 0, the indeterminate form 0 ÷ 0.
Step 5: Your Turn
Practice makes it stick.
The Cancel
Problem 1 of 2
Limit of (x² − 9) ÷ (x − 3) as x approaches 3?
The Special Limit
Problem 2 of 2
Limit of sin x ÷ x as x approaches 0?
Open the Limit
1 of 8
Limit of (x² − 4) ÷ (x − 2) as x approaches 2?
2 of 8
Limit of (x² − 25) ÷ (x − 5) as x approaches 5?
3 of 8
Limit of (x³ − 8) ÷ (x − 2) as x approaches 2?
4 of 8
Limit of (1 − cos x) ÷ x as x approaches 0?
5 of 8
Limit of 5x + 2 as x approaches 3?
6 of 8
g is trapped between −x² and x² near 0. What is its limit there?
7 of 8
Sort each form by what it tells you.
Tap something to move it.
- Empty
- Empty
8 of 8
Limit of (x² − 16) ÷ (x + 4) as x approaches −4?
Step 6: Quick Check
Show what you know.
Question 1 of 2
Limit of (x² − 1) ÷ (x − 1) as x approaches 1?
Question 2 of 2
What does the Squeeze Theorem require?
What You Learned
- 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.
- sin x/x → 1 and (1 − cos x)/x → 0; match the inside to the denominator to use them.