Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/38

If a variable $\scriptstyle {v}$ ultimately becomes and remains in numerical value greater than any assigned positive number however large, we say $\scriptstyle {v}$, in numerical value, increases without limit, or $\scriptstyle {v}$ becomes infinitely great,^[1] and write

$\scriptstyle {\operatorname {limit\;} v=\infty }$, or, $\scriptstyle {v\doteq \infty }$.

Infinity ($\scriptstyle {\infty }$) 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.

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

If the independent variable $\scriptstyle {x}$ takes on any series of values such that

$\scriptstyle {\operatorname {limit\;} x=a}$,

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

$\scriptstyle {\operatorname {limit\;} f(x)=A}$,

then as a single statement this is written

$\scriptstyle {{\underset {x=a}{\operatorname {limit} }}\;f(x)=A}$,

and is read the limit of $\scriptstyle {f(x)}$, as $\scriptstyle {x}$ approaches the limit $\scriptstyle {a}$ in any manner, is $\scriptstyle {A}$.

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

$\scriptstyle {{\underset {x=a}{\operatorname {limit} }}f(x)=f(a)}$,

then $\scriptstyle {f(x)}$ is continuous for $\scriptstyle {x=a}$. The function is said to be discontinuous for $\scriptstyle {x=a}$ if this condition is not satisfied. For example, if

$\scriptstyle {{\underset {x=a}{\operatorname {limit} }}f(x)=\infty }$,

the function is discontinuous for $\scriptstyle {x=a}$.

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

1. On account of the notation used and for the sake of uniformity, the expression $\scriptstyle {v\doteq +\infty }$ is sometimes read $\scriptstyle {v}$ approaches the limit plus infinity. Similarly, $\scriptstyle {v\doteq -\infty }$ is read $\scriptstyle {v}$ approaches the limit minus infinity, and $\scriptstyle {v\doteq \infty }$ is read $\scriptstyle {v}$, 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.