Limits at $+\infty$ and $-\infty$, and infinite limits at a point

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

Limits at $+\infty$. 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

$\lim_{x \to +\infty} f(x) = L$

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

$|f(x) - L| < \varepsilon \qquad \text{for every } x \in A \text{ with } x > M .$

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

Why unboundedness is required. It plays exactly the role the limit-point condition plays in The $\varepsilon$-$\delta$ 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 $\lim_{x \to +\infty} f(x) = L$ and $\lim_{x \to +\infty} f(x) = L'$ with $L \ne L'$. Then $|L - L'| > 0$ (Basic properties of the absolute value), so $\varepsilon := |L - L'|/2 > 0$ (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 $\varepsilon$ 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

$|L - L'| = |(L - f(x)) + (f(x) - L')| \le |f(x) - L| + |f(x) - L'| < 2\varepsilon = |L - L'|$

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

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

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

when for every real $M$ there is a real $\delta > 0$ such that $f(x) > M$ for every $x \in A$ with $0 < |x - c| < \delta$; and $f(x) \to -\infty$ 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 $\lim_{x \to c} f(x) = +\infty$. 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 $+\infty$ and to $-\infty$ for sequences and by Conventions: $\sup \emptyset$, 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 $\pm\infty$.

Combined forms. Let $A$ be not bounded above and $f : A \to \mathbb{R}$. We write $f(x) \to +\infty$ as $x \to +\infty$ 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 $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ the sets $A \cap N^{*}_\delta(c)$ 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 $A \cap (-\infty, c)$ and $A \cap (c, \infty)$ gives, for instance, $f(x) \to +\infty$ as $x \to c^{+}$, meaning $f^{+}(x) \to +\infty$ as $x \to c$ for the restriction $f^{+}$ of $f$ to $A \cap (c, \infty)$, 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 $\overline{\mathbb{R}}$ of The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined exists in this library and is the right home for $\limsup$; 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.