INDETERMINATE FORMS 
209 
Since (2) is true, we have: 
(5) fja + h) = f'jX) 
K F(a + h) F'(X') 
If, now, these derivatives exist and are continuous at the point 
x = a, too; and if F'(a)=£ 0, then* 
(6) 
F(x) o F'(X') F\a) 
For example, 
(7) 
lim -XZ 
x=M> X 
(8) 
Rut if, as in the case of 
1 — cos x 
= -= 1. 
1 
a = 0, 
/'(a) and F\a) both vanish, equation (6) breaks down, nor can we do 
anything with (5) since we do not know how Xand X' vary relatively 
to each other. This case can be dealt with as follows. 
Generalized Law of the Mean. If fix) and F(x) are continu 
ous throughout the interval a ^ x ^ b and each has a derivative at all 
interior points of the interval, and. if moreover, the derivative F'(x) 
does not vanish within the interval; then, for some value x = X within 
this interval, 
(9) 
m-m = rm 
F(b)— F(a) F'(X)’ 
a < X < b. 
* In this case the result can be obtained at once, since 
fix) _ f(g + h)-f(a) / F(a + h)- F(a) 
F{x) h / h * 
and the limit of the right-hand side is seen to be f'ia)/F'ia). This is known as 
“ l’Hospital’s Rule,” dating from 1G!)6. 
The limit is also called the “ true value” of the “indeterminate form” 
fix)/Fix) for x — a. Both terms are based on a false conception. In the early 
days of the Calculus mathematicians thought of the fraction as really having a 
value when x — a, only the value cannot be computed because the form of the 
fraction eludes us. This is wrong. Division by 0 is not a process which we 
define in Algebra. It is convenient, however, to retain the term indeterminate 
form as applying to such expressions as the above and others considered in this 
chapter, which for a certain value of the independent variable cease to have a 
meaning, but which approach a limit when the independent variable converges 
toward the exceptional value.