Key takeaways
- lim(x→a) f(x) = L means f(x) gets as close as you like to L as x approaches a.
- For any polynomial, lim(x→a) f(x) = f(a): just substitute.
- If substitution gives 0/0, you must simplify first.
- Use brackets for negative numbers: (−2)² = 4.
In this guide
What is a limit?
A limit describes where a function is heading. We write lim(x→a) f(x) = L. It means: as x gets closer and closer to a (from both sides), f(x) gets closer and closer to L.
Picture walking along the graph towards x = a without stepping on it. If the height of the graph is heading to a single value from both the left and the right, that value is the limit. The function doesn't even need to be defined at a itself.
Why do limits matter?
Limits are the foundation of calculus. A derivative is the limit of slopes over smaller and smaller gaps. An integral is the limit of sums of thinner and thinner strips. If you understand limits, the rest of calculus makes sense.
Limits also handle expressions that can't be computed directly. Take (x² − 4)/(x − 2). At x = 2 it gives 0/0, which means nothing. But look at nearby values:
| x | 1.9 | 1.99 | 2.01 | 2.1 |
|---|---|---|---|---|
| (x² − 4)/(x − 2) | 3.9 | 3.99 | 4.01 | 4.1 |
The function is undefined at x = 2, yet it is clearly heading to 4. That's exactly what a limit captures.
The formula: direct substitution
Polynomial graphs are smooth, unbroken curves with no holes or jumps. That means they are continuous everywhere, so the limit at any point equals the function's value there.
| Symbol | What it means | Example |
|---|---|---|
| f(x) | The polynomial xⁿ + b·x + c used by this calculator | x³ − 2x + 1 |
| n | The power (a whole number ≥ 0 for a true polynomial) | 3 |
| b | The number multiplying x | −2 |
| c | The constant term | 1 |
| a | The value x approaches | 2 |
How to find the limit step by step
- 1Check it's a polynomial. Or at least continuous at a, with no division by zero.
- 2Substitute. Replace every x with the number a.
- 3Follow the order of operations. Powers first, then multiplying, then adding.
- 4Read off the answer. The number you get is the limit.
Now try your own polynomial below. Choose n, b, c and the point a to see the substitution worked out.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting brackets with negative numbers. If a = −2, then
a² = (−2)² = 4, not −4. - Thinking the limit and f(a) are always the same. That is true only for continuous functions. See the continuity checker.
- Doing operations in the wrong order. Work out the power before multiplying and adding.
- Checking only one side. A limit exists only if the left and right sides agree.
Tips and tricks
- Sums and products: the limit of a sum is the sum of the limits, and the same goes for products.
- Polynomials never jump, so you can trust substitution every single time with them.
- Famous exception: sin(x)/x at 0 gives 0/0, but its limit is 1. See the squeeze theorem.
Where you'll use it in real life
- Calculus: every derivative and integral is defined as a limit.
- Physics: instantaneous speed is the limit of average speed over shorter and shorter times.
- Engineering: checking how a system behaves as a setting approaches a critical value.
- Finance: continuous compounding is a limit of compounding more and more often.
Quick summary
A limit is the value a function heads towards as x approaches a. For polynomials, which are continuous everywhere, just substitute a and simplify. If you ever get 0/0, simplify first. Use the calculator above to check any polynomial limit step by step.
Worked example
Your homework asks for the limit of f(x) = x³ − 2x + 1 as x approaches 2. This fits the form xⁿ + bx + c with n = 3, b = −2 and c = 1, and the limit point is a = 2.
- 01f(x) = x^3 + -2x + 1
- 02lim[x→2] f(x) — polynomial: continuous everywhere
- 03= f(2) (direct substitution)
- 04= 2^3 + -2(2) + 1
- 05= 8 + -4 + 1
- 06= 5
These numbers are pre-filled in the calculator above.
Frequently asked questions
Can I always just plug in the number to find a limit?
Only when the function is continuous at that point, such as polynomials. If plugging in gives 0/0 or a number divided by 0, you need another method.
What is the limit of x² − x + 3 as x approaches 2?
Substitute x = 2: 4 − 2 + 3 = 5. So the limit is 5.
What does 0/0 mean when finding a limit?
It is called an indeterminate form. It does not mean the limit is 0 or undefined; it means you must simplify, factor or use L'Hôpital's rule to find the real value.
Does a function need to be defined at a to have a limit there?
No. The limit only cares about values near a, not at a. (x² − 4)/(x − 2) is undefined at 2 but its limit there is 4.
