Key takeaways
- lim (x→c) f(x) = L means f(x) gets as close to L as you like when x gets close to c.
- A limit exists only if the left-hand and right-hand limits are equal.
- A function is continuous at c if f(c) exists, the limit exists, and they are equal.
- For 0/0, try factoring first, then L'Hôpital's rule: lim f/g = lim f′/g′.
In this guide
What is a limit?
A limit describes the value a function gets closer and closer to as the input gets closer to some number. It does not care what happens exactly at that number, only what happens near it.
Read it as: the limit of f(x) as x approaches c is L. It means you can make f(x) as close to L as you want by taking x close enough to c.
Why do limits matter?
Limits are the foundation of all of calculus. A derivative is a limit of slopes. An integral is a limit of sums of thin rectangles. Without limits, neither would have a precise meaning.
Calculus is built on one question: what happens as we get closer and closer?
Key rules and theorems
| Tool | When to use it | Key idea |
|---|---|---|
| Direct substitution | f is a polynomial or is continuous at c | Just put x = c in |
| Factoring | Substitution gives 0/0 | Cancel the common factor, then substitute |
| Squeeze theorem | f is trapped between two simpler functions | If both outer limits equal L, so does f |
| L'Hôpital's rule | 0/0 or ∞/∞ | Differentiate top and bottom separately |
L'Hôpital's rule, written in plain notation:
How to evaluate a limit step by step
- 1Try direct substitution. Put x = c into the function. If you get a normal number, that is the limit.
- 2If you get 0/0, simplify. Factor, cancel, or rationalise, then substitute again.
- 3Still stuck? Use L'Hôpital's rule for 0/0 or ∞/∞: differentiate top and bottom separately.
- 4Check both sides if the function is split into pieces or has a jump.
- 5Use the squeeze theorem for wiggly functions like
x²·sin(1/x).
The limit calculator evaluates polynomial limits of the form xⁿ + bx + c by direct substitution, and the L'Hôpital's rule calculator solves lim (x→a) (xⁿ − aⁿ)/(x − a), which always equals n·aⁿ⁻¹. For example, lim (x→2) (x⁴ − 16)/(x − 2) = 4 × 2³ = 32.
Continuity
A function is continuous if you can draw its graph without lifting your pencil. Formally, f is continuous at x = c when three things are true:
- 1f(c) is defined. The function has a value at c.
- 2The limit exists.
lim (x→c) f(x)is a real number. - 3They match.
lim (x→c) f(x) = f(c).
If any condition fails, there is a break: a hole, a jump or a vertical asymptote. For example, f(x) = 1/x is not continuous at 0, because f(0) is not defined. The continuity checker tests all three conditions for powers of x.
The squeeze theorem
If g(x) ≤ f(x) ≤ h(x) near c, and g and h both approach L, then f must approach L too. The famous example is lim (x→0) sin(x)/x = 1. Try it with the squeeze theorem calculator: at h = 0.01, sin(0.01)/0.01 ≈ 0.99998, very close to 1.
Common mistakes to avoid
- Using the quotient rule in L'Hôpital's rule. Differentiate the top and bottom separately, not as a fraction.
- Using L'Hôpital's rule when it doesn't apply. Check for 0/0 or ∞/∞ first.
- Checking only one side. At a jump, the left and right limits differ, so the limit does not exist.
- Confusing the limit with the value. A function can have a limit at c even if f(c) is undefined.
Tips and tricks
- Polynomials are always continuous, so direct substitution always works for them.
- Remember two classics:
lim (x→0) sin(x)/x = 1andlim (x→0) (eˣ − 1)/x = 1. - Limits at infinity: for a fraction of polynomials, compare the highest powers on top and bottom.
Where you'll use it in real life
- Calculus: every derivative and integral is defined by a limit.
- Physics: instant speed is the limit of average speed over shorter and shorter times.
- Finance: continuous compound interest is a limit, and it leads to the number e.
- Engineering: checking that a system behaves smoothly with no sudden jumps.
Quick summary
A limit is the value a function approaches, written lim (x→c) f(x) = L. Try substitution first; if you get 0/0, factor or use L'Hôpital's rule; for trapped functions, use the squeeze theorem. A function is continuous at c when f(c) exists, the limit exists, and they are equal. The free calculators linked above show each step.
Frequently asked questions
What is an indeterminate form?
It is an expression like 0/0 or infinity/infinity that has no single value on its own. You need to simplify, factor or use L’Hôpital’s rule to find the limit.
How does the Squeeze Theorem work?
If g(x) ≤ f(x) ≤ h(x) near a point, and g(x) and h(x) both approach the same limit L, then f(x) must approach L too. It is squeezed between them.
Can a limit exist if the function is undefined at that point?
Yes. The limit only looks at values near the point. For example, (x² − 9)/(x − 3) is undefined at x = 3, but its limit there is 6.
What makes a function discontinuous?
A break in the graph: a hole where the function is undefined, a jump where the left and right limits differ, or a vertical asymptote where the values shoot off to infinity.
