Limits of Functions

Sequences let $x$ approach a target through a discrete list $x_1,x_2,\dots$ The limit of a function poses the same question for a continuous variable: as $x$ ranges over a set $S$ and moves toward a point $c$, does $f(x)$ settle on one value $L$? The definition is the sequence definition with past index $N$ replaced by within distance $\delta$ of $c$, and almost everything about functional limits is recovered by turning the continuous approach back into sequences.

Two points need care before the definition works. First, $x$ can only approach $c$ if $S$ actually has points arbitrarily near $c$; otherwise the condition is vacuous. Second, the value $f(c)$ is irrelevant — $f$ need not even be defined at $c$, and when it is, ${\lim }_{x\to c}f(x)$ can differ from $f(c)$. A limit describes $f$ near $c$ while ignoring $c$ itself.

Cluster points

A cluster point need not belong to $S$. The word isolates exactly the points at which approach within $S$ is possible.

• The set $\{1/n:n\in \mathbb{N}\}$ has the single cluster point $0$, which is not in the set.
• The cluster points of $(0,1)$ are all of $[0,1]$; the endpoints are cluster points not contained in the open interval.
• Every real number is a cluster point of $\mathbb{Q}$, by density of the rationals.
• The set $[0,1)\cup \{2\}$ has cluster-point set $[0,1]$; the isolated point $2$ is not a cluster point.
• $\mathbb{N}$ has no cluster points in $\mathbb{R}$: around any real, only finitely many integers lie within distance $1$.

The definition ties directly to convergent sequences.

The epsilon–delta definition

Read $ϵ$ as a target tolerance on the output and $\delta$ as the input window that achieves it. The clause $x\ne c$ (the $\setminus \{c\}$) is what lets the limit ignore the value at $c$. Geometrically, for every horizontal band of half-height $ϵ$ around $L$, there is a vertical strip of half-width $\delta$ around $c$ in which the graph stays inside the band.

Capping $\delta$ at $1$ first bounds the slope factor $|x+c|$; the remaining factor of $\delta$ then finishes the estimate. The same two-stage choice recurs throughout $ϵ$–$\delta$ arguments with a nonconstant slope.

Uniqueness

The cluster-point hypothesis is what supplies the point $x$; without it every $L$ would satisfy the definition vacuously.

The sequential (Heine) criterion

The Heine criterion converts functional limits into sequential limits, so the algebra of sequence limits transfers directly.

The quantifier ranges over every sequence. To prove a limit fails it is enough to produce one bad sequence, or two sequences whose images converge to different values.

The algebra of limits

Feeding the Heine criterion into the sequence limit laws produces the function limit laws with no new work. Let $c$ be a cluster point of $S$ and suppose ${\lim }_{x\to c}f(x)$ and ${\lim }_{x\to c}g(x)$ exist.⁴

• Order. If $f(x)\le g(x)$ on $S\setminus \{c\}$, then ${\lim }_{x\to c}f(x)\le {\lim }_{x\to c}g(x)$.
• Bounds. If $a\le f(x)\le b$ on $S\setminus \{c\}$, then $a\le {\lim }_{x\to c}f(x)\le b$.
• Squeeze. If $f(x)\le g(x)\le h(x)$ near $c$ and ${\lim }_{x\to c}f={\lim }_{x\to c}h$, then ${\lim }_{x\to c}g$ exists and equals the common value.
• Sum, difference, product. The limit of $f\pm g$ and of $fg$ is the sum, difference, and product of the limits.
• Quotient. If ${\lim }_{x\to c}g(x)\ne 0$ and $g\ne 0$ on $S\setminus \{c\}$, then ${\lim }_{x\to c}\frac{f(x)}{g(x)}=\frac{\lim f}{\lim g}$.
• Absolute value. ${\lim }_{x\to c}|f(x)|=|{\lim }_{x\to c}f(x)|$.

The $x\sin (1/x)$ computation above is an instance of the squeeze law: $-|x|\le x\sin (1/x)\le |x|$, with both envelopes tending to $0$.

One-sided limits

Restricting the domain to one side of $c$ gives limits that can exist even when the two-sided limit does not.

The two-sided limit exists exactly when both one-sided limits exist and agree, since $(S\cap (-\infty ,c))\cup (S\cap (c,\infty ))=S\setminus \{c\}$.

Restrictions and locality

The limit depends only on $f$ near $c$. If $A\subseteq S$ agrees with $S$ on some punctured window $(c-\alpha ,c+\alpha )$, then $c$ is a cluster point of $A$ iff it is one of $S$, and ${\lim }_{x\to c}f={\lim }_{x\to c}f{|}_A$. One-sided limits are the standard case where $A$ is a genuine one-sided slice, so the two-sided limit can fail to exist while a restricted limit exists.

Continuity is the condition ${\lim }_{x\to c}f(x)=f(c)$: the limit exists at $c$ and equals the function's value there.