The Formal Definition of a Limit

Definition Let $f$ be defined on an open interval containing $a$, except possibly at $a$, and let $L$ be a real number. Then $\lim_{x \to a} f(x) = L$ means that for every $\varepsilon > 0$ there exists $\delta > 0$ such that for all $x$,

Definition The statement has the form “for every $\varepsilon > 0$ there exists $\delta > 0$”. The order matters. The number $\delta$ is chosen after $\varepsilon$ is given, so $\delta$ may depend on $\varepsilon$ and on $a$. If the order were swapped to “there exists $\delta > 0$ such that for every $\varepsilon > 0$”, the condition would force $\lvert f(x) - L \rvert < \varepsilon$ for all $\varepsilon > 0$, that is $f(x) = L$ for every $x$ in a punctured neighborhood of $a$. That is a much stronger and different statement.

Definition The hypothesis $0 < \lvert x - a \rvert < \delta$ has two parts. The right inequality $\lvert x - a \rvert < \delta$ keeps $x$ within $\delta$ of $a$. The left inequality $0 < \lvert x - a \rvert$ says $x \neq a$. So the condition describes exactly the punctured neighborhood $(a - \delta, a) \cup (a, a + \delta)$, and the value $f(a)$ is never tested.

Definition In the definition, the implication

must hold for every $x$ in the domain of $f$ that satisfies $0 < \lvert x - a \rvert < \delta$, not merely for some of them. Under the standard convention of this course, every point of the punctured neighborhood is such an $x$ once $\delta$ is small, because $f$ is defined on an open interval around $a$ except possibly at $a$.

Definition A number $a$ is an accumulation point of a set $D$ if every interval $(a - \delta, a + \delta)$ with $\delta > 0$ contains a point of $D$ other than $a$. The point $a$ itself may or may not belong to $D$. Some advanced texts extend the definition of $\lim_{x \to a} f(x) = L$ to a function with any domain $D$ by testing only inputs $x \in D$, and then they take limits only at accumulation points of $D$. Under the convention of this course this condition holds automatically, since $D$ contains a whole punctured neighborhood of $a$.

Definition A proof of $\lim_{x \to a} f(x) = L$ needs a rule that turns each $\varepsilon > 0$ into some $\delta > 0$ satisfying the definition. A standard method is to factor $\lvert f(x) - L \rvert$ as $\lvert x - a \rvert$ times another factor, bound that factor by first requiring $\delta \le 1$ (or some other fixed number), and then take $\delta$ to be the minimum of the two requirements. Any smaller positive $\delta$ also works.

Definition The statement $\lim_{x \to a} f(x) = L$ is false exactly when there exists $\varepsilon > 0$ such that for every $\delta > 0$ there is an $x$ with

To negate, each “for every” becomes “there exists”, each “there exists” becomes “for every”, and the implication “if $0 < \lvert x - a \rvert < \delta$ then $\lvert f(x) - L \rvert < \varepsilon$” becomes ”$0 < \lvert x - a \rvert < \delta$ and $\lvert f(x) - L \rvert \ge \varepsilon$”.