
If a variable v ultimately becomes and remains in numerical value
greater than any assigned positive number however large, we say v,
in numerical value, increases without limit, or v becomes infinitely great,[*]
and write

limit v = $$,  or,  v $$ $$.

Infinity ($$) 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.

17. Limiting value of a function. Given a function f(x).

If the independent variable x takes on any series of values such that

limit x = a,

and at the same time the dependent variable f(x) takes on a series of
corresponding values such that

limit f(x) = A,

then as a single statement this is written

limit x=a f(x)=A,

and is read the limit of f(x), as x approaches the limit a in any manner,
is A.

18. Continuous and discontinuous functions. A function f(x) is said
to be 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

limit x=a f(x)=f(a),

then f(x) is continuous for x=a.

The function is said to be discontinuous for x=a if this condition
is not satisfied. For example, if

limit x=a f(x)=$$,

the function is discontinuous for x=a.

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

* On account of the notation used and for the sake of uniformity, the expression
v $$ +$$ is sometimes read v approaches the limit plus infinity. Similarly, v $$ -$$ is read
v approaches the limit minus infinity, and v $$ $$ is read 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
infinity is not a number at all. 
