\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \newcommand{\qed} { {\scriptstyle \Box} } \require{physics} \require{ams} \require{mathtools} \)

Limit laws for functions, proposition

Combining the rudimentary definitions of arithmetic operations on functions with the unwieldy definition of function limits gives us the limit laws of functions.

As words, they are as follows:

Let \( X \) be a subset of \( \mathbb{R} \), let \( E \) be a subset of \( X \), let \( x_0 \) be [...] of \( E \) and let \( f : X \to \mathbb{R} \) and \( g : X \to \mathbb{R} \) be functions. Suppose that \( f \) has a limit \( L \) at \( x_0 \) in \( E \), and \( g \) has a limit \( M \) at [...] in \( E \), then:

• \( f+g \) has a limit \( L + M \) at \( x_0 \) in \( E \)
• \( f-g \) has a limit \( L - M \) at \( x_0 \) in \( E \)
• \( \max(f, g) \) has a limit \( \max(L, M) \) at \( x_0 \) in \( E \)
• \( \min(f, g) \) has a limit \( \min(L, M) \) at \( x_0 \) in \( E \)
• \( fg \) has a limit \( LM \) at \( x_0 \) in \( E \)
• \( cf \) has a limit \( cL \) at \( x_0 \) in \( E \) (assuming \( c \) is a real number)
• \( f/g \) has a limit \( L / M \) at \( x_0 \) in \( E \) (assuming \( g(x) \ne 0 \text{ for all } x \in E \) and \( M \ne 0 \))