CHAPTER V.

EVALUATION OF INDETERMINATE FUNCTIONS.

Indeterminateness of explicit Functions of a single Variable.

63. Suppose that $ (x) =

f(x)
F(x)'

and that, when a particular value x is assigned to x, f(x) and F(x) both become zero: the value of (x) will, for such a value of x, present itself under the indeterminate form. We proceed to investigate a rule which is often useful for the determination of the true value of (x).

x and dx being supposed to be replaced by x, and Sx, in the expressions for

SF(x)
Ex

which are generally functions of x and dx. Passing to the limit, when dx, becomes less than any assignable quantity, we have

it being supposed that, in the expressions for the functions ƒ'(x)

and F(x), x, is substituted for x. In other words, for any value of x which makes f(x) = 0 and F(x) = 0, the value of

If, for this same value of x, ƒ'(x) = 0 and F"(x) = 0; then, by the application of the same principle, we see that, for this value of x,

or

and so on.

If f(x) and F(x) are the lowest differential coefficients of f(x) and F(x), of which both do not vanish for the particular value x, of x, then the true value of ☀ (x) will be

ƒ'(x) = log a . a* - log b . b* = log a log b, when x =

= 0:

It would have been however more simple, observing that

sin =1, when x = 1, to have sought the value of

2

The value of p(x) may be obtained also, and in fact more simply, without the aid of differentials: thus