Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero

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 $\mathbb{R}$), let $f,g:A\to \mathbb{R}$ and let $\alpha \in \mathbb{R}$. Suppose the limits of $f$ and of $g$ at $c$ exist, and write $L:={\lim }_{x\to c}f(x)$ and $M:={\lim }_{x\to c}g(x)$ (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:

1. the limit of $f+g$ at $c$ exists, and ${\lim }_{x\to c}(f+g)(x)={\lim }_{x\to c}f(x)+{\lim }_{x\to c}g(x)=L+M;$
2. the limit of $\alpha f$ at $c$ exists, and ${\lim }_{x\to c}(\alpha f)(x)=\alpha {\lim }_{x\to c}f(x)=\alpha L;$
3. the limit of $fg$ at $c$ exists, and ${\lim }_{x\to c}(fg)(x)=({\lim }_{x\to c}f(x))({\lim }_{x\to c}g(x))=LM;$
4. if $M\mathrm{\ne }0$, then, writing $A_0:=\{x\in A:g(x)\mathrm{\ne }0\}$, the point $c$ is a limit point of $A_0$, the quotient $f\mathrm{/}g$ is defined on $A_0$ by $(f\mathrm{/}g)(x)=f(x)\mathrm{/}g(x)$, the limit of $(f\mathrm{/}g){\mathrm{\mid }}_{A_0}$ at $c$ exists, and ${\lim }_{x\to c}(f\mathrm{/}g){\mathrm{\mid }}_{A_0}(x)=\frac{{\lim }_{x\to c}f(x)}{{\lim }_{x\to c}g(x)}=\frac{L}{M}\mathrm{.}$

Each equation asserts two things at once: that the limit on the left exists, and that it has the stated value. Both are proved. The symbols denote by At a limit point of the domain a function has at most one limit.

Everything below is proved directly from $\epsilon$ and $\delta$. No sequence is constructed and no choice principle is used, so all four claims are theorems of ZF. Passing through Heine criterion: ${\lim }_{x\to c}f(x)=L$ iff $f(x_k)\to L$ for every sequence in $A\setminus \{c\}$ converging to $c$ instead would import the countable choice spent in that theorem's converse direction, for no gain; see The sequence-to-$\epsilon$ direction of the Heine criterion uses countable choice for $\mathbb{R}$, and where this library records that cost.

Why the quotient is stated on $A_0$. The function $f\mathrm{/}g$ is simply not defined where $g$ vanishes, and $g$ may well vanish at points of $A$ arbitrarily far from $c$; restricting to $A_0$ is therefore forced. That this restriction still has $c$ as a limit point, so that the limit there means anything at all, is the last claim of If ${\lim }_{x\to c}f(x)=L\mathrm{\ne }0$ then $\mathrm{\mid }f\mathrm{\mid }>\mathrm{\mid }L\mathrm{\mid }\mathrm{/}2$ on a punctured neighbourhood of $c$; in particular if $L>0$ then $f>L\mathrm{/}2>0$ there. The sequential analogue Algebra of limits: sums, scalar multiples, products and quotients needs the corresponding hypothesis in the form "the denominator sequence is nonzero at every index".

• The product estimate in one line. The identity $fg-LM=f(g-M)+M(f-L)$ turns the problem into two products, one with a factor that is merely bounded near $c$ (that is $f$, and If $f$ has a finite limit at $c$ then $f$ is bounded on some punctured neighbourhood of $c$ is what bounds it) and one with a constant factor. The two constants $K^{\mathrm{\prime }}$ and $\mathrm{\mid }M\mathrm{\mid }+1$ are used in place of $K$ and $\mathrm{\mid }M\mathrm{\mid }$ only so that they are strictly positive and may be divided by; that is the sole reason for adding $1$.

• The reciprocal estimate in one line. The identity $1\mathrm{/}g-1\mathrm{/}M=(M-g)\mathrm{/}(gM)$ turns the problem into a numerator that is small and a denominator that must be kept away from $0$; the lower bound $\mathrm{\mid }g\mathrm{\mid }>\mathrm{\mid }M\mathrm{\mid }\mathrm{/}2$ from If ${\lim }_{x\to c}f(x)=L\mathrm{\ne }0$ then $\mathrm{\mid }f\mathrm{\mid }>\mathrm{\mid }L\mathrm{\mid }\mathrm{/}2$ on a punctured neighbourhood of $c$; in particular if $L>0$ then $f>L\mathrm{/}2>0$ there does exactly that, and gives the working factor $2\mathrm{/}\mathrm{\mid }M{\mathrm{\mid }}^2$.

• Nothing here extends to $\pm \mathrm{\infty }$. The statement is about a finite limit point $c$ and finite values $L,M$; Limits at $+\mathrm{\infty }$ and $-\mathrm{\infty }$, and infinite limits at a point introduces limits at and to infinity, but no algebra of such limits is proved in this library, and none may be assumed. The companion page's limit at $+\mathrm{\infty }$ is computed by a direct estimate for precisely that reason.

• The sequential analogue is Algebra of limits: sums, scalar multiples, products and quotients. Neither implies the other for free: this theorem is about a function on a subset of $\mathbb{R}$ and is proved from $\epsilon$ and $\delta$; that one is about sequences.