Calculus › Limits
Limits
A limit asks: as x sneaks up on some value, what does f(x) head toward? It's the foundation of everything else in calculus.
Sneaking up on a limit
f(x) = (x² − 1) / (x − 1). What is f(1)?
From the left
| x | f(x) |
|---|---|
| 0 | 1.0000 |
From the right
| x | f(x) |
|---|---|
| 2 | 3.0000 |
What's a limit, really?
A limit is the value a function gets arbitrarily close to as the input gets arbitrarily close to some point. The function might not even be defined at that point — that's fine. The limit is about the behaviour near the point.
Why we need them
Take f(x) = (x² − 1) / (x − 1). At x = 1, you'd be dividing by zero. But for any x close to 1 (but not equal to 1), the expression simplifies to x + 1. So as x sneaks up on 1, f(x) sneaks up on 2.
Notation
limx→1 f(x) = 2. Read: "the limit of f(x) as x approaches 1 is 2."Left and right
For a limit to exist, both sides have to agree. If the function jumps at the point (different values from left and right), the two-sided limit doesn't exist.