calculus foundations
Limits ask what happens as a value approaches a point.
Definition
The limit of a function describes the value a function approaches as its input gets closer and closer to some point. It is a central idea in calculus because it captures behavior near a point without requiring exact value at that point.
Example
Consider the function \( f(x) = \frac{x^2 - 1}{x - 1} \). For \( x \neq 1 \), this simplifies to \( x + 1 \). As \( x \to 1 \), the function approaches \( 2 \), even though the original expression is undefined at \( x = 1 \).
Continuity
A function is continuous at a point if the limit exists, the function value exists, and they are equal. Intuitively, a continuous function can be drawn without lifting the pen.
Why it matters
Limits and continuity are the foundation of derivatives and integrals. They allow us to study motion, rates of change, and accumulation in a precise mathematical way.
Key idea
A limit is about approach, not necessarily equality at the point itself. This idea is what gives calculus its power: it allows us to understand behavior near values that may not be directly defined there.