Limits of Real Functions

Throughout, $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property)) with its order and absolute value (Order on the reals).

Let $A\subseteq \mathbb{R}$, let $f:A\to \mathbb{R}$, let $c\in \mathbb{R}$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), and let $L\in \mathbb{R}$. We say that $f(x)$ tends to $L$ as $x$ tends to $c$, and write

${\lim }_{x\to c}f(x)=L,$

when

$(\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in A)[0<\mathrm{\mid }x-c\mathrm{\mid }<\delta ⟹\mathrm{\mid }f(x)-L\mathrm{\mid }<\epsilon ],$

where $\epsilon$ and $\delta$ range over the positive reals.

In the language of neighbourhoods (The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$) the condition reads: for every real $\epsilon >0$ there is a real $\delta >0$ with

$f(A\cap N_{\delta }^{\ast }(c))\subseteq N_{\epsilon }(L),$

$N_{\delta }^{\ast }(c)=\{y:0<\mathrm{\mid }y-c\mathrm{\mid }<\delta \}$ being the punctured $\delta$-neighbourhood of $c$ and $N_{\epsilon }(L)=(L-\epsilon ,L+\epsilon )$ the open interval of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length. The two forms agree because $\mathrm{\mid }f(x)-L\mathrm{\mid }<\epsilon$ says exactly $f(x)\in N_{\epsilon }(L)$, and $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$ says exactly $x\in N_{\delta }^{\ast }(c)$.

Three features of this definition are load bearing, not decoration.

1. $c$ is required to be a limit point of $A$. By Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$ that says every punctured neighbourhood of $c$ meets $A$, so for every $\delta >0$ the set $A\cap N_{\delta }^{\ast }(c)$ over which the implication quantifies is nonempty. Drop the requirement and the implication can be satisfied vacuously by every real $L$ at once, which is exactly what FALSE: a function has at most one limit at every point of its domain, isolated points included records. At a point of $A$ that is not a limit point of $A$ — an isolated point — the symbol ${\lim }_{x\to c}f(x)$ is therefore not defined in this library.

2. $c\in A$ is not required. A limit point of $A$ need not belong to $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), and the definition never evaluates $f$ at $c$. This is what allows a limit to be taken at a point where the function is not defined at all, as at $0$ for $x\mapsto x\psi (1\mathrm{/}x)$.

3. The value $f(c)$, when it exists, is irrelevant. The hypothesis $0<\mathrm{\mid }x-c\mathrm{\mid }$ excludes $x=c$ from the quantifier, so changing $f$ at the single point $c$ changes nothing. Equality of the limit with the value is an extra condition, not a consequence: FALSE: ${\lim }_{x\to c}f(x)=f(c)$ whenever both sides exist.

The notation presumes uniqueness. Writing ${\lim }_{x\to c}f(x)=L$ treats the left-hand side as a name for a single real number, which is legitimate only because at a limit point at most one $L$ can satisfy the displayed condition. That obligation is discharged by At a limit point of the domain a function has at most one limit ↗, recorded in this item's justified_by. As with $\sup S$ (Conventions: $\sup \mathrm{\varnothing }$, unbounded sets, and the extended reals) and ${\lim }_k{x}_k$ (A sequence has at most one limit), the symbol is written only for a function already known to have a limit at $c$.

Real and rational $\epsilon$ define the same relation. Above, $\epsilon$ and $\delta$ range over the positive reals. Restricting either quantifier to the positive rationals gives the same relation: every positive rational is a positive real, and below every positive real lies a positive rational (The rationals embed densely in the reals), so an $\epsilon$-condition verified for all positive rationals is verified for an arbitrary positive real $\eta$ by running it at a rational $\epsilon$ with $0<\epsilon <\eta$, and a $\delta$ produced as a real may be shrunk to a rational one below it. This is the passage sanctioned in the remarks of Sequences of reals: bounded, eventually, frequently, tails, subsequences, and it is what lets this definition be compared with Limits and Cauchy sequences of reals, whose $\epsilon$ is rational, in Heine criterion: ${\lim }_{x\to c}f(x)=L$ iff $f(x_k)\to L$ for every sequence in $A\setminus \{c\}$ converging to $c$.

• Terminology. Limit point here is a property of the set $A$ and the point $c$, in the sense of Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$; it has nothing to do with subsequential limits (Subsequential limit of a real sequence, and the subsequential limit set), and the distinction is the one that item records.

• Why the punctured condition, and not $\mathrm{\mid }x-c\mathrm{\mid }<\delta$. With the unpunctured condition the definition would force $f$ to be defined at $c$ and would force $\mathrm{\mid }f(c)-L\mathrm{\mid }<\epsilon$ for every $\epsilon$, that is, $L=f(c)$. The resulting notion is continuity at $c$, a strictly stronger condition, and conflating the two is the error catalogued in FALSE: ${\lim }_{x\to c}f(x)=f(c)$ whenever both sides exist.

• One-sided and infinite variants. Restricting the domain to one side of $c$ gives the one-sided limits of The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$; replacing the conditions on $x$ or on $f(x)$ by unboundedness conditions gives the limits at and to infinity of Limits at $+\mathrm{\infty }$ and $-\mathrm{\infty }$, and infinite limits at a point. Both are built on this definition rather than beside it.

Let $A\subseteq \mathbb{R}$, let $f:A\to \mathbb{R}$, let $c$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$) and let $L,L^{\mathrm{\prime }}\in \mathbb{R}$. If

${\lim }_{x\to c}f(x)=L\text{and}{\lim }_{x\to c}f(x)=L^{\mathrm{\prime }}$

(The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$), then $L=L^{\mathrm{\prime }}$.

A function therefore has at most one limit at a limit point of its domain, which is what licenses the notation ${\lim }_{x\to c}f(x)$ for a single real number. This lemma is recorded in the justified_by field of The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$ for exactly that reason.

The hypothesis that $c$ is a limit point is not removable. At an isolated point of the domain the same $\epsilon$-$\delta$ formula is satisfied vacuously by every real at once, which is the content of FALSE: a function has at most one limit at every point of its domain, isolated points included.

• Where each hypothesis is spent. The limit conditions are used only in step 5.1, and the limit-point hypothesis only in step 4.1, to produce a single point $x$ of the domain near $c$ at which both estimates can be read. Without such a point the two estimates never meet and nothing forces $L=L^{\mathrm{\prime }}$; that is the whole mechanism, and it is the reason FALSE: a function has at most one limit at every point of its domain, isolated points included is false.

• The sequential analogue is A sequence has at most one limit, proved by the same two-estimates-at-one-index argument. Neither statement uses any choice principle.

• One-sided limits inherit this. By The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$ a one-sided limit is the limit of a restriction of $f$, so applying this lemma to that restriction gives uniqueness there too; nothing has to be reproved.

Let $A\subseteq \mathbb{R}$ and let $c$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$).

1. Locality. Let $f,g:A\to \mathbb{R}$ and $L\in \mathbb{R}$, and suppose there is a real $\eta >0$ with $f(x)=g(x)$ for every $x\in A$ satisfying $0<\mathrm{\mid }x-c\mathrm{\mid }<\eta$. Then ${\lim }_{x\to c}f(x)=L⟺{\lim }_{x\to c}g(x)=L$ (The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$).

2. Restriction. Let $B\subseteq A$ with $c$ a limit point of $B$, let $f:A\to \mathbb{R}$ and suppose ${\lim }_{x\to c}f(x)=L$. Then $c$ is a limit point of $A$ as well, and ${\lim }_{x\to c}f{\mathrm{\mid }}_B(x)=L$, where $f{\mathrm{\mid }}_B:B\to \mathbb{R}$ is the restriction of $f$.

So the limit at $c$ sees only the values of $f$ on an arbitrarily small punctured neighbourhood of $c$, and it survives shrinking the domain, provided the smaller domain still accumulates at $c$. Together with At a limit point of the domain a function has at most one limit this is what makes the phrase the limit at $c$ a local notion.

The converse of claim 2 is false in general: a restriction may have a limit where the function has none, as the one-sided limits of the sign function on the companion page show.

• What claim 1 is used for. It is the licence to modify a function outside a punctured neighbourhood of $c$, or at $c$ itself, without changing the limit; the change at $c$ alone is already invisible to The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$, since the condition $0<\mathrm{\mid }x-c\mathrm{\mid }$ excludes that point.

• What claim 2 is used for. It is the step that lets a statement proved on $A$ be transported to a smaller domain: the one-sided limits of The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$ are exactly limits of restrictions, and the quotient case of Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero is proved on the smaller domain where the denominator does not vanish.

• Both claims are choice free. Only the $\epsilon$-$\delta$ definition is used; no sequence is constructed anywhere in the proof.

Let $A\subseteq \mathbb{R}$, let $f:A\to \mathbb{R}$ and let $c\in \mathbb{R}$. Put

$A^{-}:=A\cap (-\mathrm{\infty },c),A^{+}:=A\cap (c,\mathrm{\infty })$

(Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and write $f^{-}:=f{\mathrm{\mid }}_{A^{-}}$ and $f^{+}:=f{\mathrm{\mid }}_{A^{+}}$ for the restrictions of $f$ to those sets.

Right limit. Suppose $c$ is a limit point of $A^{+}$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$). For $L\in \mathbb{R}$ we write

${\lim }_{x\to c^{+}}f(x)=L:⟺{\lim }_{x\to c}{f}^{+}(x)=L$

in the sense of The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$. Written out: for every real $\epsilon >0$ there is a real $\delta >0$ such that

$\mathrm{\mid }f(x)-L\mathrm{\mid }<\epsilon \text{for every }x\in A\text{ with }c<x<c+\delta \mathrm{.}$

Left limit. Suppose $c$ is a limit point of $A^{-}$. For $L\in \mathbb{R}$ we write ${\lim }_{x\to c^{-}}f(x)=L$ when ${\lim }_{x\to c}{f}^{-}(x)=L$; written out, for every real $\epsilon >0$ there is a real $\delta >0$ with $\mathrm{\mid }f(x)-L\mathrm{\mid }<\epsilon$ for every $x\in A$ with $c-\delta <x<c$.

The written-out forms agree with the definitions. For $x\in A^{+}$ the two conditions $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$ and $c<x<c+\delta$ are the same: $x>c$ gives $x-c>0$, so $\mathrm{\mid }x-c\mathrm{\mid }=x-c$ and $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$ reads $0<x-c<\delta$ (Basic properties of the absolute value). Symmetrically on the left, where $x<c$ gives $\mathrm{\mid }x-c\mathrm{\mid }=c-x$.

Well-posedness is inherited, not reproved. A one-sided limit is a limit, namely the limit of a restriction, so:

• Uniqueness. At most one $L$ can occur, by At a limit point of the domain a function has at most one limit applied to $f^{+}$ on the domain $A^{+}$ (respectively to $f^{-}$ on $A^{-}$), which is legitimate exactly because $c$ was required to be a limit point of that set. This is what makes the notation ${\lim }_{x\to c^{+}}f(x)$ denote a single real.
• Locality and restriction. Both claims of The limit at $c$ depends only on the restriction of $f$ to a punctured neighbourhood of $c$, and passes to any subset of the domain having $c$ as a limit point apply verbatim to $f^{+}$ and $f^{-}$.

When the symbols are defined. If $c$ is not a limit point of $A^{+}$ — for instance if $A$ contains no point to the right of $c$, or only points bounded away from $c$ on that side — then ${\lim }_{x\to c^{+}}f(x)$ is not defined here, for the reason given in The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$: the $\epsilon$-$\delta$ condition would be satisfied vacuously by every real at once. The same applies on the left.

• Neither one-sided limit requires $c\in A$, and neither looks at $f(c)$. Both properties are inherited from The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$, since $c\mathrm{\notin }{A}^{-}\cup A^{+}$: the point $c$ belongs to neither $(-\mathrm{\infty },c)$ nor $(c,\mathrm{\infty })$.

• The two one-sided limits and the two-sided limit. When $c$ is a limit point of both $A^{-}$ and $A^{+}$, the two-sided limit exists exactly when both one-sided limits exist and agree, and then all three coincide: If $c$ is a limit point of the domain from both sides, the limit exists iff both one-sided limits exist and agree. When $c$ is a limit point of only one of the two sets, that one-sided limit and the two-sided limit are the same condition, again by claim 2 of The limit at $c$ depends only on the restriction of $f$ to a punctured neighbourhood of $c$, and passes to any subset of the domain having $c$ as a limit point together with the observation that $A$ and that one side have the same points in a small enough punctured neighbourhood of $c$.

• Notation. Some texts write $f(c+)$ and $f(c-)$ for these values. This library writes only ${\lim }_{x\to c^{+}}f(x)$ and ${\lim }_{x\to c^{-}}f(x)$, because the shorter notation looks like an evaluation of $f$ and these quantities are not values of $f$: they are defined without reference to $f(c)$, which may not even exist.

Throughout, $+\mathrm{\infty }$ and $-\mathrm{\infty }$ are abbreviations and not real numbers, exactly as in Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length and Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$. Every phrase below is a single abbreviation for a displayed condition on reals, and no arithmetic is ever performed with the symbols.

Limits at $+\mathrm{\infty }$. Let $A\subseteq \mathbb{R}$ be not bounded above (Lower bound, bounded below, bounded set), let $f:A\to \mathbb{R}$ and let $L\in \mathbb{R}$. We write

${\lim }_{x\to +\mathrm{\infty }}f(x)=L$

when for every real $\epsilon >0$ there is a real $M$ such that

$\mathrm{\mid }f(x)-L\mathrm{\mid }<\epsilon \text{for every }x\in A\text{ with }x>M\mathrm{.}$

Limits at $-\mathrm{\infty }$. Let $A$ be not bounded below. We write ${\lim }_{x\to -\mathrm{\infty }}f(x)=L$ when for every real $\epsilon >0$ there is a real $M$ with $\mathrm{\mid }f(x)-L\mathrm{\mid }<\epsilon$ for every $x\in A$ with $x<M$.

Why unboundedness is required. It plays exactly the role the limit-point condition plays in The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$. Saying that $A$ is not bounded above says that no real is an upper bound of $A$, that is, that for every real $M$ there is $x\in A$ with $x>M$ (Lower bound, bounded below, bounded set, Complete ordered field (least-upper-bound property)); so the set over which the condition quantifies is never empty and the condition is never vacuous. Without the hypothesis every real $L$ would satisfy it and the notation would not denote.

Uniqueness, proved here. Suppose $A$ is not bounded above and ${\lim }_{x\to +\mathrm{\infty }}f(x)=L$ and ${\lim }_{x\to +\mathrm{\infty }}f(x)=L^{\mathrm{\prime }}$ with $L\mathrm{\ne }{L}^{\mathrm{\prime }}$. Then $\mathrm{\mid }L-L^{\mathrm{\prime }}\mathrm{\mid }>0$ (Basic properties of the absolute value), so $\epsilon :=\mathrm{\mid }L-L^{\mathrm{\prime }}\mathrm{\mid }\mathrm{/}2>0$ (The multiplicative identity is positive, Inverses of positives are positive, and reciprocation reverses order, Sign rules for products and monotonicity of multiplication). Choose reals $M_1,M_2$ witnessing the two conditions at this $\epsilon$ and let $M$ be the larger of them, the order being total. Since $A$ is not bounded above there is $x\in A$ with $x>M$, hence with $x>M_1$ and $x>M_2$, and then

$\mathrm{\mid }L-L^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }(L-f(x))+(f(x)-L^{\mathrm{\prime }})\mathrm{\mid }\le \mathrm{\mid }f(x)-L\mathrm{\mid }+\mathrm{\mid }f(x)-L^{\mathrm{\prime }}\mathrm{\mid }<2\epsilon =\mathrm{\mid }L-L^{\mathrm{\prime }}\mathrm{\mid }$

(The triangle inequality, Basic properties of the absolute value, Order is preserved by adding a constant and by adding inequalities), which trichotomy forbids. So $L=L^{\mathrm{\prime }}$, and the notation ${\lim }_{x\to +\mathrm{\infty }}f(x)$ denotes a single real. The same four lines, with the inequalities on $x$ reversed, give uniqueness at $-\mathrm{\infty }$.

Infinite limits at a point. Let $A\subseteq \mathbb{R}$, let $c$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$) and let $f:A\to \mathbb{R}$. We write

$f(x)\to +\mathrm{\infty }\text{ as }x\to c$

when for every real $M$ there is a real $\delta >0$ such that $f(x)>M$ for every $x\in A$ with $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$; and $f(x)\to -\mathrm{\infty }$ as $x\to c$ when for every real $M$ there is a real $\delta >0$ with $f(x)<M$ for every such $x$.

This library does not write ${\lim }_{x\to c}f(x)=+\mathrm{\infty }$. The right-hand side would not be an element of $\mathbb{R}$, and writing the equation would silently move the discussion into the extended real line, a structure that is not a field. That is the convention already fixed by Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$ for sequences and by Conventions: $\sup \mathrm{\varnothing }$, unbounded sets, and the extended reals for suprema, and it is kept here. In particular none of the rules of Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero may be applied to a function tending to $\pm \mathrm{\infty }$.

Combined forms. Let $A$ be not bounded above and $f:A\to \mathbb{R}$. We write $f(x)\to +\mathrm{\infty }$ as $x\to +\mathrm{\infty }$ when for every real $N$ there is a real $M$ with $f(x)>N$ for every $x\in A$ with $x>M$. The other forms are obtained the same way, by pairing one of the two conditions on $x$ (unbounded above, unbounded below) with one of the two conditions on $f(x)$ (above every real, below every real); each is again a single abbreviation for the displayed condition, and none of them is an equation.

• These are the same definition with a different notion of "near". In The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$ the sets $A\cap N_{\delta }^{\ast }(c)$ shrink to $c$; here the sets $\{x\in A:x>M\}$ shrink towards being unbounded above. The limit-point hypothesis and the unboundedness hypothesis play the same role: each says the relevant sets are never empty.

• One-sided infinite limits. Combining this definition with The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$ gives, for instance, $f(x)\to +\mathrm{\infty }$ as $x\to c^{+}$, meaning $f^{+}(x)\to +\mathrm{\infty }$ as $x\to c$ for the restriction $f^{+}$ of $f$ to $A\cap (c,\mathrm{\infty })$, provided $c$ is a limit point of that set. Nothing new has to be defined.

• The extended reals are not needed on these pages. The extended line $\bar{\mathbb{R}}$ of The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined exists in this library and is the right home for $\mathrm{lim\hspace{0.17em}sup}$; it is deliberately not used here, because every statement above is a statement about reals and quantifiers, and introducing a second ordered structure would oblige every later algebraic step to say which structure it is working in.

Let $A\subseteq \mathbb{R}$, let $f:A\to \mathbb{R}$, let $c$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$) and let $L\in \mathbb{R}$. The following are equivalent.

1. ${\lim }_{x\to c}f(x)=L$ (The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$).
2. For every sequence $(x_k{)}_{k\in \mathbb{N}}$ with $x_k\in A$ and $x_k\mathrm{\ne }c$ for every $k$, and $x_k\to c$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals), the sequence $(f(x_k){)}_{k\in \mathbb{N}}$ converges to $L$.

The two directions do not cost the same. The implication from 1 to 2 is proved in ZF: the sequence is handed to the proof, and nothing is selected. The implication from 2 to 1, as proved below, invokes the axiom of countable choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) exactly once, at step 3.2, to select one bad point from each of countably many nonempty sets. What this library does and does not claim about that cost is recorded in The sequence-to-$\epsilon$ direction of the Heine criterion uses countable choice for $\mathbb{R}$, and where this library records that cost; the same asymmetry appears, for the same reason, in A point lies in the closure of $A\subseteq \mathbb{R}$ iff some sequence in $A$ converges to it, so a subset of $\mathbb{R}$ is closed iff it is sequentially closed.

Because of this, the results on this page that can be proved directly from $\epsilon$ and $\delta$ — the algebra of limits, order preservation, the squeeze theorem, composition — are proved that way, and not through this criterion. What the criterion is for is the transfer of sequential results to functions, and above all the negative use recorded in A function has no limit at $c$ as soon as two sequences in $A\setminus \{c\}$ tending to $c$ give different limits of the values, which needs only the choice-free direction.