CHAPTER XL.
Limiting Values And Vanishing Fractions.
456. It will be convenient here to introduce a phraseology and notation which the student will frequently meet with in his mathematical reading.
457. Functions. An expression which involves any quantity, as 2, and whose value is dependent on that of 2, is called a function of x. Functions of x are usually denoted by symbols of the form f(x), f'(x), F(x), \phi(x), and read “the f function of x,” “the f’ function of 2,” etc.
Thus the equation y = f(x) may be considered equivalent to a statement that any change made in the value of x will produce a consequent change in y, and vice versa. The quantities x and y are called variables, and are further distinguished as the independent variable and the dependent variable.
An independent variable is a quantity which may have any value we choose to assign to it, and the corresponding dependent variable has its value determined as soon as the value of the independent variable is known.
458. Definition. If y= f(x), and if when x approaches a value a, the function f(x) can be made to differ by as little as we please from a fixed quantity b, then b is called the limit of y when x=a.
For instance, if S denote the sum of n terms of the series 1 1 1 41 5 at 2 yi ra +++; then S=
