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

14 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, and write

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.

Given a function $$\scriptstyle{f(x)}$$.

If the independent variable $$\scriptstyle{x}$$ takes on any series of values such thatand at the same time the dependent variable $$\scriptstyle{f(x)}$$ takes on a series of corresponding values such thatthen as a single statement this is written and is read the limit of $$\scriptstyle{f(x)}$$, as $$\scriptstyle{x}$$ approaches the limit $$\scriptstyle{a}$$ in any manner, is $$\scriptstyle{A}$$.

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, ifthen $$\scriptstyle{f(x)}$$ 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 the function is discontinuous for $$\scriptstyle{x=a}$$.

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