Limits and Continuity

The Approach to a Limit

Calculus is built on the limit: the value a function approaches as its input gets close to a point, even if the function is never evaluated there. We write $\lim_{x \to a} f(x) = L$ to say as $x$ approaches $a$, $f(x)$ approaches $L$.

A one-sided limit checks the approach from just the left ($x \to a^-$) or right ($x \to a^+$). A two-sided limit exists if and only if both one-sided limits agree.

Continuity and Pitfalls

Continuity captures having no breaks in a graph. A function $f$ is continuous at $a$ when three conditions hold: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$.

Polynomials are continuous everywhere. A hole, jump, or vertical asymptote creates a discontinuity.

Common pitfall: Confusing the limit of a function with its value at that point. $\lim_{x \to a} f(x)$ can exist even when $f(a)$ is undefined. The limit ignores the point $a$ and looks only at nearby behavior. Continuity is the special case where limit and value agree.