1. What a Limit Really Tells Us
A limit describes the value a function gets closer and closer to as the input moves towards a particular point. The input may never actually reach that point — what matters is the behaviour nearby.
Limits help capture the idea of “approaching” a value, even if the function is not defined at that point.
2. Why Limits Are Needed
Some functions behave strangely at certain points: they may jump, blow up, or be undefined. Limits allow us to understand the behaviour around those points. This idea forms the gateway to differentiation and continuity.
3. Basic Notation
Limits have a standard notation that shows both the function and the point approached.
3.1. Notation
\lim_{x \to a} f(x)
This reads as: “the value of f(x) as x approaches a”.
4. Understanding Limits Through Values
The easiest way to understand limits is to observe what happens to the outputs when x moves closer to a value.
4.1. Example
Consider:
f(x) = x^2
\lim_{x \to 2} x^2 = 4
As x gets closer to 2 (1.9, 1.99, 1.999…), x² gets closer to 4.
5. Limits Even When the Function Is Not Defined
A limit depends on the approach, not the exact value at the point. So a function may be undefined at x = a and still have a limit at x = a.
5.1. Example
f(x) = \frac{x^2 - 4}{x - 2}
This expression is undefined at x = 2, but when simplified:
f(x) = x + 2, \; x \ne 2
As x approaches 2, f(x) approaches 4. So:
\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4
6. Graphical View of Limits
On a graph, a limit captures where the y-values head as x gets closer to a point from both sides. Even if there is a hole in the graph, the limit can still exist.
6.1. Key Idea
A limit exists if the left and right sides approach the same value.
7. Behaviour Near the Point Matters
To find a limit, we never plug in values far from the point. We only care about values close to it.
7.1. Example
For \( \lim_{x \to 3} (x+1) \), the values of the function near x = 3 (like 2.9, 3.01) matter, not values like 10 or -5.
8. What It Means When a Limit Exists
If the limit exists at a point, the function behaves predictably around that point. Even if the function jumps or breaks elsewhere, near that point the behaviour is stable.