Always substitute first. For a continuous function the limit is the value. If substitution gives 0/0, the top and bottom share a factor: factor and cancel, rationalise, or clear the fractions, then substitute again.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Work out (x² − 16) ÷ (x − 4) and x + 4 at x = 3.9, 3.99, 4.01 and 4.1 — every division here comes out exactly. What do the two rules do, and what does that say about x = 4?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
The indeterminate form
0/0 is not an answer. It reports that the top and bottom share a factor, which must be removed before the limit appears. (x² − 9)/(x − 3) at 3: 0/0, but it is (x + 3) in disguise, limit 6.
Factor and cancel
(x² − 5x + 6)/(x − 2) as x → 2: factor to (x − 2)(x − 3)/(x − 2) = x − 3. Limit: 2 − 3 = −1. Canceling changes the function at exactly one point, the one point a limit ignores.
When a root is in the way
Multiply by the conjugate. (√x − 3)/(x − 9) as x → 9: times (√x + 3)/(√x + 3) gives (x − 9)/((x − 9)(√x + 3)) = 1/(√x + 3). Limit: 1/6.
Stacked fractions
(1/x − 1/2)/(x − 2) as x → 2: the top is (2 − x)/(2x), so the whole is (2 − x)/(2x(x − 2)) = −1/(2x). Limit: −1/4.
Not all zeros are equal
A non-zero number over zero is not indeterminate. 1/(x − 2) as x → 2 gives 1/0: the values run away, and the limit does not exist.
A numeric check
After the algebra, test a nearby x. (√(0.01 + 4) − 2)/0.01 = 0.2498, close to 1/4. One substitution confirms the whole page.
Step 2: Try It Yourself
Tap and try it out.
The limit of (x² − 5x + 6)/(x² − 4) as x → 2
- 1x = 2: (4 − 10 + 6)/(4 − 4) = 0/0indeterminate: a shared factor is hiding
Step 0 of 4
- Point(1, 2)
Step 3: In Real Life
A formula that divides by zero
Average cost per item is total cost over items, and at zero items it is 0/0. Factoring the formula shows what the cost per item approaches for the very first unit. Substitution fails; algebra answers.
Step 4: Watch an Example
One step at a time.
Watch Priya Clear a Shared Bracket
A fresh fraction, (x² − 2x − 8) ÷ (x² − 4x), as x approaches 4. Priya substitutes first, reads what comes back, factors both halves, and finishes with a value from each side.
- Step 1
Substitution comes first, always. At x = 4 the top is 16 − 8 − 8 = 0 and the bottom is 16 − 16 = 0, so the fraction hands back 0 ÷ 0. That is not an answer. It is a report that both halves vanish at 4, which means each of them carries a factor of (x − 4), and finding those two factors is the whole job.
Watch Leo Clear a Root
The same kind of problem with a root in the way: the limit of (√(4x + 9) − 5) ÷ (x − 4) as x approaches 4. Leo starts it, and you finish the last three steps.
- Step 1
Substitution is the first move here too. At x = 4, 4x + 9 is 25, whose root is 5, so the top is 5 − 5 = 0, and the bottom is 4 − 4 = 0. Another 0 ÷ 0 — but nothing on this page factors, because the root is in the way.
Watch Ana Tell Two Look-Alikes Apart
Two fractions with the same bottom, (x − 5) ÷ (x² − 25) and (x + 5) ÷ (x² − 25), both as x approaches 5. One sign apart, and two different kinds of answer.
- Step 1
Two fractions, differing by one sign, both asked at x = 5. Nothing about how they look decides which is which; substitution does, and it is the first move for each of them.
Step 5: Your Turn
Practice makes it stick.
The Cost of the First Poster
Problem 1 of 2
A print shop’s total cost for x posters is 4x² + 12x dollars. The cost per poster is that total divided by x. At x = 0 the formula gives 0 ÷ 0. What does the cost per poster approach as x approaches 0, in dollars?
dollars
Dollars an Hour
Problem 2 of 2
A market stall’s takings after t hours are 5t² dollars. Its average takings per hour between hour 2 and hour t are (5t² − 20) ÷ (t − 2). What does that average approach as t approaches 2, in dollars per hour?
dollars per hour
Substitute, Then Clear the Way
Substitution first in every one. What it hands back decides what the next move is.
1 of 5
What is the limit of 2x² − 3x + 1 as x approaches 4?
2 of 5
What is the limit of (x² − 7x + 10) ÷ (x − 5) as x approaches 5?
3 of 5
What is the limit of (x² − x − 12) ÷ (x² − 16) as x approaches 4? Type it as a decimal.
4 of 5
What is the limit of (√(x + 3) − 2) ÷ (x − 1) as x approaches 1? Type it as a decimal.
5 of 5
What is the limit of (1/x − 1/5) ÷ (x − 5) as x approaches 5? Type it as a decimal.
Whose Step?
One line on each page is a slip. Find it, then put it right.
1 of 6
Maya worked out the limit of (x² − 4x + 4) ÷ (x − 2) as x approaches 2. One line is a slip. Which?
Maya’s working for the limit of (x² − 4x + 4)/(x − 2) as x → 2
- 1x = 2: (4 − 8 + 4)/(2 − 2) = 0/0
- 20/0 = 1, since anything divided by itself is 1
- Answerthe limit is 1
2 of 6
What is the limit of (x² − 4x + 4) ÷ (x − 2) as x approaches 2?
3 of 6
Sam worked out the limit of (√(x + 7) − 4) ÷ (x − 9) as x approaches 9. One line is a slip. Which?
Sam’s working for the limit of (√(x + 7) − 4)/(x − 9) as x → 9
- 1x = 9: (4 − 4)/(9 − 9) = 0/0
- 2(√(x + 7) − 4)(√(x + 7) + 4) = (x + 7) − 16 = x − 9
- 3so the fraction is (x − 9)/(x − 9) = 1, for x ≠ 9
- Answerthe limit is 1
4 of 6
What is the limit of (√(x + 7) − 4) ÷ (x − 9) as x approaches 9? Type it as a decimal.
5 of 6
Noor worked out the limit of (1/x − 1/4) ÷ (x − 4) as x approaches 4. One line is a slip. Which?
Noor’s working for the limit of (1/x − 1/4)/(x − 4) as x → 4
- 1x = 4: (1/4 − 1/4)/(4 − 4) = 0/0
- 21/x − 1/4 = (4 − x)/(4x)
- 3(4 − x)/(4x(x − 4)) = 1/(4x), for x ≠ 4
- Answerthe limit is 1/16 = 0.0625
6 of 6
What is the limit of (1/x − 1/4) ÷ (x − 4) as x approaches 4? Type it as a decimal.
Pick the Move
Decide which road the work takes. Neither of these has to be carried all the way to a number.
1 of 2
For the limit of (x² + 5) ÷ (x − 1) as x approaches 1, which first move is the right one?
2 of 2
For the limit of (x³ − 27) ÷ (x − 3) as x approaches 3, which first move opens it?
Same Limit?
Substitute into each one first, and only then decide how much more work it needs.
1 of 2
Select every one of these whose limit is 6.
2 of 2
Select every one of these whose limit is exactly the value substitution hands back.
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
Substituting gives 0 ÷ 0. What does that mean?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
For which number k does the limit of (x² + kx − 12) ÷ (x − 3) as x approaches 3 exist as a number?
Stretch 2 of 2
A fraction has bottom (x − 5) and a top of the form x² + bx + c. Choose b and c so that substituting 5 gives 0 ÷ 0 and the limit as x approaches 5 is 8. Type the top, expanded.
What You Learned
- 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.
- Confirm the algebra with one nearby value.
For the grown-up
The misconception this lesson targets is reading 0 ÷ 0 as a number — usually 0, sometimes 1, because "anything over itself is one" — instead of as a report that the work has not started. It is neither an answer nor a failure: it says both halves vanish at the point, so both carry the same factor, and the limit is whatever survives once that factor is taken away. Its cousin is the opposite error, treating a non-zero number over zero as the same kind of message and hunting for a factor that is not there. Substituting first, and then reading which of the two came back, decides everything that follows, and one nearby value confirms the answer in a single line.