Let \(S \subset \R\text{,}\) \(c \in \R\text{,}\) \(f \colon S \to \R\) be increasing, and \(g \colon S \to \R\) be decreasing. If \(c\) is a cluster point of \(S \cap (-\infty,c)\text{,}\) then
\begin{equation*}
\lim_{x \to c^-} f(x) = \sup \{ f(x) : x < c, x \in S \}
\quad \text{and} \quad
\lim_{x \to c^-} g(x) = \inf \{ g(x) : x < c, x \in S \} .
\end{equation*}
If \(c\) is a cluster point of \(S \cap (c,\infty)\text{,}\) then
\begin{equation*}
\lim_{x \to c^+} f(x) = \inf \{ f(x) : x > c, x \in S \}
\quad \text{and} \quad
\lim_{x \to c^+} g(x) = \sup \{ g(x) : x > c, x \in S \} .
\end{equation*}
If \(\infty\) is a cluster point of \(S\text{,}\) then
\begin{equation*}
\lim_{x \to \infty} f(x) = \sup \{ f(x) : x \in S \}
\quad \text{and} \quad
\lim_{x \to \infty} g(x) = \inf \{ g(x) : x \in S \} .
\end{equation*}
If \(-\infty\) is a cluster point of \(S\text{,}\) then
\begin{equation*}
\lim_{x \to -\infty} f(x) = \inf \{ f(x) : x \in S \}
\quad \text{and} \quad
\lim_{x \to -\infty} g(x) = \sup \{ g(x) : x \in S \} .
\end{equation*}