A limit asks where a function is heading as x approaches a number, not what it does there. (x² − 1)/(x − 1) is undefined at x = 1, yet as x approaches 1 the value approaches 2.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A function is given by two rules: f(x) = 2x − 4 for every x except 3, and f(3) = 7. Work f out at x = 2.9, 2.99, 3.01 and 3.1, and put those four numbers beside f(3). Which description fits what you have?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Watch it approach
x = 0.9, 0.99, 0.999 give 1.9, 1.99, 1.999. x = 1.1, 1.01, 1.001 give 2.1, 2.01, 2.001. From both sides the values close in on 2. That is what the limit records.
Writing it
lim as x → 1 of (x² − 1)/(x − 1) = 2. Read: the limit, as x approaches 1, is 2. The arrow is the whole idea: approaching, never arriving.
Limit against value
The function value at 1 does not exist. The limit at 1 is 2. Two different questions, and this lesson is about keeping them apart.
Why this matters so much
The derivative is exactly this: a quantity that is undefined at the point, 0/0, and has a limit there. All of calculus rests on being able to read that limit.
One-sided limits
Approaching from the left, x → 1⁻, and from the right, x → 1⁺, can give different answers. For a step function jumping from 3 to 5 at x = 2, the left limit is 3 and the right limit is 5.
When there is no limit
If the two one-sided limits differ, the limit does not exist. If the values run away, as 1/x does near 0, it does not exist either. A limit is one number the function settles toward.
Step 2: Try It Yourself
Tap and try it out.
f(x) = x + 1 for x < 2, and f(x) = 2x for x ≥ 2. Does the limit at x = 2 exist?
- 1from the left: x + 1 → 2 + 1 = 3x → 2⁻ uses the first piece
Step 0 of 4
- Point(2, 0.50)
Step 3: In Real Life
A speedometer
A car’s speed at one instant is a limit: distance over a shorter and shorter interval. No interval is zero long, yet the needle points at a number. That number is where the ratio is heading.
Step 4: Watch an Example
One step at a time.
Watch Maya Find the Limit of (2x² − 18)/(x − 3) at 3
A fresh fraction with a hole, opened by factoring, and checked against its neighbours at the end.
- Step 1
Substituting 3 straight away gives (2 × 9 − 18)/(3 − 3), which is 0/0. That is not a number and it is not an answer. It is a signal: the top and the bottom both vanish at 3, which happens only when both of them contain the factor (x − 3).
Watch Leo Test a Join, Then Finish It Yourself
f(x) = x² for x < 1, and f(x) = 3x − 2 for x ≥ 1. Leo asks whether there is a limit at x = 1. He works the left side; the right side, the verdict, and one change to the rule are yours.
- Step 1
The rule changes at x = 1, so the two sides have to be asked separately: below 1 the function is x², and at 1 and above it is 3x − 2. A limit at 1 is a question about neighbours, and the neighbours on the two sides are covered by different formulas.
Watch Ana Take Two Roads to One Limit
The limit of (x² − 5x + 6)/(x − 2) as x approaches 2, found once with a table of neighbours and once with algebra. Both roads reach the same number, and only one of them settles it.
- Step 1
Ana takes the first road, a table of neighbours. At x = 1.99 the top is 3.9601 − 9.95 + 6 = 0.0101 and the bottom is −0.01, giving −1.01. At x = 2.01 the top is 4.0401 − 10.05 + 6 = −0.0099 and the bottom is 0.01, giving −0.99. From both sides the values close in on −1.
Step 5: Your Turn
Practice makes it stick.
The Blank Cell
Problem 1 of 2
A price table was built in a spreadsheet from the formula p(x) = (x² − 100)/(x − 10) dollars, where x is the number of boxes ordered. Every row is filled except the row for 10 boxes, where the formula comes to 0 ÷ 0 and the cell is blank. What price is the table heading toward at 10 boxes, in dollars?
dollars
The Free Shipping Line
Problem 2 of 2
An online shop charges $6 shipping on any order below $50, and nothing at all on an order of $50 or more. A shopper keeps adding items and the order total climbs toward $50. Which description of the shipping charge at $50 is right?
Where Is It Heading?
Some of these finish in one line and some need algebra first. Find out which kind you are holding before you commit to a road.
1 of 5
What is the limit of 4x − 7 as x approaches 3?
2 of 5
What is the limit of (x² − 49)/(x − 7) as x approaches 7?
3 of 5
f(x) = x + 5 for x < 2, and f(x) = 4x for x ≥ 2. What is the limit as x approaches 2 from the left?
4 of 5
f(x) = x² − 1 for every x except 4, and f(4) = 20. What is the limit as x approaches 4?
5 of 5
What is the limit of (3x² − 12)/(x − 2) as x approaches 2?
Whose Step?
One line on each page is a slip. Find it, then put it right.
1 of 4
Priya worked out the limit of (x² − 3x − 10)/(x − 5) as x approaches 5. One line is a slip. Which?
Priya’s working for the limit of (x² − 3x − 10)/(x − 5) as x → 5
- 1x = 5 gives (25 − 15 − 10)/(5 − 5) = 0/0
- 2x² − 3x − 10 = (x − 5)(x − 2)
- 3(x − 5)(x − 2)/(x − 5) = x − 2, for x ≠ 5
- Answerlimit = 5 − 2 = 3
2 of 4
Factored and cancelled correctly, what does (x² − 3x − 10)/(x − 5) equal for every x except 5? Type the expression.
3 of 4
Dev tested whether f has a limit at x = 4, where f(x) = x + 6 for x < 4 and f(x) = 3x for x ≥ 4. One line is a slip. Which?
Dev’s test for a limit at x = 4
- 1from the left: x + 6 → 4 + 6 = 10
- 2from the right: 3x → 3 × 4 = 12
- 310 and 12 differ, so the value at 4 decides it: f(4) = 12
- Answerlimit = 12
4 of 4
Read correctly, what is the limit of f at x = 4?
Pick the Move
Choose the move that answers the question. Neither of these needs working all the way through.
1 of 2
The limit of (x² + x − 12)/(x − 3) as x approaches 3 is wanted. What is the first move?
2 of 2
f(x) = 7 − x for x < 6, and f(x) = x − 5 for x ≥ 6. Which move settles whether f has a limit at x = 6?
Same Limit?
Judge each one on its neighbours. Select every entry that fits.
1 of 2
Select every limit below whose value is 4.
2 of 2
Select every case where the limit at the point and the function’s value at that point are the same number.
Step 6: Quick Check
Show what you know.
Question 1 of 2
Limit of (x² − 36)/(x − 6) as x approaches 6?
Question 2 of 2
What does a limit describe?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
f(x) = 5x + k for x < 2, and f(x) = x² + 9 for x ≥ 2. For which number k does f have a limit at x = 2?
Stretch 2 of 2
For exactly one number b does (x² + bx − 24)/(x − 3) come to 0 ÷ 0 at x = 3. Find that b, then type the limit of the fraction as x approaches 3.
What You Learned
- A limit is where a function heads, not what it equals.
- A function need not be defined at a point to have a limit there.
- If the two one-sided limits differ, the limit does not exist.
- 0/0 is not an answer. It is a signal that algebra is needed first.
For the grown-up
The misconception this lesson targets is that a limit is a slower way of saying a function’s value: that where the value is missing there is nothing to find, and that where the value is present the limit has to match it. A student holding it answers every substitution correctly and reads every hole and every jump as a misprint in the question. It costs nothing for a while and then costs everything, because a derivative is a limit taken at exactly the place where the quotient reads 0 ÷ 0 — so a student who hears 0 ÷ 0 as "no answer" has ruled out the rest of the subject before it starts — and the slip that travels with it is letting the value at the point break the tie when the two sides disagree, which is what a jump looks like from the inside. The habit that settles both is cheap: substitute first and read what comes back, then work the neighbours on each side separately, and only afterwards ask what the function does at the point, as a separate question with its own answer.