Limits and Continuity

Approaching Without Arriving

Calculus is built on the limit — the value a function approaches as its input approaches some point, even if the function is never actually evaluated there. We write $\lim_{x \to a} f(x) = L$ to say "as $x$ gets arbitrarily close to $a$, $f(x)$ gets arbitrarily close to $L$." Crucially, the limit is about the approach, not the value at $a$ — the function might be undefined at $a$ yet still have a limit there.

A subtlety is one-sided limits: the value approached from the left ($x \to a^-$) may differ from the value approached from the right ($x \to a^+$). The (two-sided) limit exists only when both sides agree. A jump in the graph is exactly a case where the two one-sided limits differ, so no limit exists.

Continuity captures "no breaks in the graph." A function $f$ is continuous at $a$ when three things all hold:

1. $f(a)$ is defined,
2. $\lim_{x \to a} f(x)$ exists, and
3. the two are equal: $\lim_{x \to a} f(x) = f(a)$.

Intuitively, you can draw a continuous function without lifting your pen. Polynomials are continuous everywhere; a function with a hole, jump, or vertical asymptote is discontinuous there. Continuity matters because the powerful theorems of calculus (and reliable numerical computation) depend on it.

Common pitfall: confusing the limit of a function at a point with its value at that point. $\lim_{x \to a} f(x)$ can exist even when $f(a)$ is undefined or different — the limit ignores the single point $a$ and looks only at nearby behavior. Continuity is precisely the special case where the limit and the value agree; assuming they always agree erases the whole point of a limit.