Defining Limits and Using Limit Notation

Plugging in works fine when a function is well-behaved, but plenty of functions are undefined, broken, or weird at exactly the point you care about. Limits let you describe where a function is headed as x sneaks up on a value, even if the function never actually gets there or does something strange when it arrives. This matters because calculus is built on questions about behavior near a point, not just at a point: slopes of curves, instantaneous rates, areas under curves. All of these start by asking what happens as you get arbitrarily close, not what happens exactly at one spot.

Limit notation, written lim(x→a) f(x) = L, says that as x gets closer and closer to a from both sides, f(x) gets closer and closer to L. It makes no claim about f(a) itself. The function might be undefined there, might have a hole, or might even be defined to equal something completely different. The limit only reads the trend of nearby values, never the value sitting at the point itself. That's why you can find a limit at a point where the function doesn't officially exist.

This separation between 'what the function does nearby' and 'what the function equals at the point' sets up the next idea: continuity. A function is continuous at a point exactly when those two things finally agree, when the limit equals the actual function value. Limits also set up derivatives, since a derivative is itself a limit of slopes as two points squeeze together. Getting comfortable with the idea that limits ignore the point itself is what makes both of those later ideas click.