Always substitute first. For a continuous function the limit is the value, and the work is finished.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The indeterminate form
0 ÷ 0 gives no information. It reports that a shared factor is hiding the answer, nothing more.
Three techniques
Factor and cancel, rationalise with a conjugate, or clear a compound fraction. One of these usually opens it.
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.
The classic use
x² sin(1/x) oscillates wildly near zero, but it is trapped between −x² and x². Both go to zero, so it must too.
The special limits
The limit of sin x ÷ x as x approaches zero is 1, and (1 − cos x) ÷ x approaches 0. Both appear constantly on the exam.
What an indeterminate form means
0/0 does not mean the limit fails to exist. It means substitution has not settled the question, because different functions of that form have different limits. More work is required, not a different conclusion.
Factor, conjugate, simplify
Factor and cancel; multiply by a conjugate to clear a radical; combine a complex fraction. Each removes the source of the zero denominator and reveals the limit that was there.
L'Hôpital's rule and its conditions
For 0/0 or ∞/∞ the limit of f/g equals the limit of f′/g′, when that exists. Applying it to a form that is not indeterminate gives a wrong answer, so the form must be verified and stated first.
The squeeze theorem
If f is trapped between two functions sharing a limit, it shares that limit too. It is how the limit of sin(x)/x is established, which in turn is what makes the derivative of sine come out as cosine.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 2)
Step 3: 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 4: 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 5: 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.