% [** F1: Using \lim for limit several times on this page]

If a variable $v$ ultimately becomes and remains in numerical value
greater than any assigned positive number however large, we say $v$,
\emph{in numerical value, increases without limit}, or \emph{$v$~becomes infinitely great},\footnote%
{On account of the notation used and for the sake of uniformity, the expression
$v \doteq +\infty$ is sometimes read \emph{$v$~approaches the limit plus infinity}. Similarly, $v \doteq -\infty$ is read
\emph{$v$~approaches the limit minus infinity}, and $v \doteq \infty$ is read \emph{$v$,~in numerical value, approaches
the limit infinity}.

While the above notation is convenient to use in this connection, the student must not
forget that infinity is not a limit in the sense in which we defined a limit on
p.~11, for % [** F1: Hard-coded page number]
infinity is not a number at all.} % End of footnote; next line included for legibility
%
and write
\[
\lim v = \infty,\quad \text{or},\quad v \doteq \infty.
\]

Infinity ($\infty$) is not a number; it simply serves to characterize a
particular mode of variation of a variable by virtue of which it
increases or decreases without limit.

\paragraph{17.} \textbf{Limiting value of a function}. Given a function~$f(x)$.

If the independent variable~$x$ takes on any series of values such that
\[
\lim x = a,
\]
and at the same time the dependent variable $f(x)$ takes on a series of
corresponding values such that
\[
\lim f(x) = A,
\]
then as a single statement this is written
\[
\lim_{x=a} f(x) = A,
\]
and is read \emph{the limit of~$f(x)$, as $x$ approaches the limit~$a$ in any manner,
is~$A$}.

\paragraph{18.} \textbf{Continuous and discontinuous functions.} A function $f(x)$ is said
to be \emph{continuous for $x=a$} if the limiting value of the function when $x$
approaches the limit~$a$ in any manner is the value assigned to the
function for $x=a$. In symbols, if
\[
\lim_{x=a} f(x) = f(a),
\]
then $f(x)$ is \emph{continuous} for $x=a$.

The function is said to be \emph{discontinuous} for $x=a$ if this condition
is not satisfied. For example, if
\[
\lim_{x=a} f(x) = \infty,
\]
the function is discontinuous for $x=a$.

The attention of the student is now called to the following cases
which occur frequently.
