Limit of a function

$\lim_{x\to a} f(x) = L$: the values $f(x)$ can be made as close to $L$ as we like by taking $x$ close enough to $a$ (but not equal). Derivatives, integrals and continuity are all defined as limits.

Some of the most important quantities are of the form $0/0$: the slope $\frac{f(a+h) - f(a)}{h}$ at $h = 0$, the average speed over a zero-length interval. We cannot plug in the value, but we can ask what the expression approaches. The limit is the tool that makes "instantaneous" and "infinitely thin" rigorous.

A game between two players. The skeptic names a tolerance $\varepsilon$ around $L$ (a horizontal band); you must answer with a $\delta$ (a vertical band around $a$) such that the graph, over the vertical band minus the point $a$ itself, stays inside the horizontal band. If you can always answer, the limit is $L$. What happens exactly at $a$ is irrelevant — $f$ need not even be defined there.

Equivalently (Heine): for every sequence $x_n \to a$ with $x_n \ne a$, $f(x_n) \to L$.

1. Try substitution: if $f$ is continuous at $a$, the limit is $f(a)$.
2. If you get $0/0$, simplify: factor, rationalize, or use known limits and equivalences.
3. Still stuck: L'Hôpital's rule or a Taylor expansion.
4. To show a limit does not exist, find two sequences $x_n \to a$ with different limits of $f(x_n)$.

$\lim_{x\to 2}\frac{x^2 - 4}{x - 2}$ is $0/0$, but $\frac{x^2-4}{x-2} = x + 2$ for $x \ne 2$, so the limit is $4$. Numerically, evaluating at $x = 2 + 10^{-12}$ in floating point gives a value off by about $10^{-4}$: the subtraction $x^2 - 4$ cancels almost all significant digits.

Every concept from here on — continuity, derivative, integral, series, convergence of algorithms — is a limit. And the example above shows the computational twist: a limit is a statement about exact arithmetic, and a computer that tries to approach it literally (tiny $h$) runs into rounding error.