Limits and Continuity

The idea of a limit

Calculus is the mathematics of change, and almost every idea in it rests on a single foundation: the limit. A limit describes the value a function approaches as its input approaches some point — regardless of what happens exactly at that point. Informally, we write

\[\lim_{x \to a} f(x) = L\]

to mean that \(f(x)\) can be made as close to \(L\) as we like by taking \(x\) sufficiently close to \(a\) (but not equal to \(a\)) ¹.

The parenthetical “but not equal to \(a\)” is the whole point. A limit asks where the function is heading, not where it is. A function can have a limit at a point where it is undefined — which is exactly what makes limits powerful enough to define the derivative later.

The formal (epsilon–delta) definition

The intuitive phrasing — “as close as we like” — is made precise with the epsilon–delta definition. We say \(\lim_{x \to a} f(x) = L\) if

\[\forall\, \varepsilon > 0 \;\; \exists\, \delta > 0 \;\; \text{such that} \;\; 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon.\]

Read it as a challenge–response: for any tolerance \(\varepsilon\) on the output, we can find a tolerance \(\delta\) on the input that guarantees \(f(x)\) lands within \(\varepsilon\) of \(L\). If such a \(\delta\) always exists, the limit is \(L\) ¹.

One-sided limits

Sometimes a function approaches different values from the left and the right. The one-sided limits capture this:

\[\lim_{x \to a^{-}} f(x) \quad\text{(from below)}, \qquad \lim_{x \to a^{+}} f(x) \quad\text{(from above)}.\]

The two-sided limit \(\lim_{x \to a} f(x)\) exists if and only if both one-sided limits exist and are equal. When they differ — as at a jump — the limit does not exist.

Limit laws

Limits respect arithmetic, which is what makes them computable. If \(\lim_{x\to a} f(x)\) and \(\lim_{x\to a} g(x)\) both exist, then

\[\lim_{x\to a}\big[f(x) + g(x)\big] = \lim_{x\to a} f(x) + \lim_{x\to a} g(x),\]

and similarly for differences, products, and quotients (provided the denominator’s limit is nonzero) ¹. For polynomials and other continuous functions these laws reduce limit evaluation to simple substitution — the interesting cases are precisely those where substitution fails, such as the indeterminate form \(\tfrac{0}{0}\).

Continuity

A function is continuous at \(a\) when the limit exists, the function is defined, and the two agree:

\[\lim_{x \to a} f(x) = f(a).\]

This compact equation bundles three requirements: \(f(a)\) exists, \(\lim_{x\to a} f(x)\) exists, and they are equal ¹. A function continuous at every point of an interval is continuous on that interval — informally, its graph can be drawn without lifting the pen.

When continuity fails, it does so in characteristic ways:

a removable discontinuity (a single missing or misplaced point, a “hole”),
a jump discontinuity (the one-sided limits disagree), and
an infinite discontinuity (the function grows without bound, as at a vertical asymptote).

Why engineers care

Continuity and limits are not abstractions for their own sake. The derivative — the rate of change central to every dynamics, circuits, and control problem — is defined as a limit, and it exists only where a function is sufficiently smooth. The next lesson builds the derivative directly on the limit machinery introduced here.