330 Functions, graphs, and numbers
and the desired formula (2) follows from this. The generalized mean-value
theorem has numerous applications, including the following one. Since x* lies
between a and x, (2) implies that
(6) limf(x) - f(a)= limf W
x-.a g(x) - g(a) x-+a g, (x)provided the limit on the right exists.
For the special case in which f(a) = g_(q) = 0, (6) reduces to the famous and
useful L'Hopital formula
(7) limf(x)= lim f, x)
x-+a g(x) x-.a g'(x)which, like (6), is correct when the limit on the right exists.t In this case the
quotient f(x)/g(x) is said to be "an indeterminate form of the form 0/0 when
x = a." The formula (7), which gives a method for finding limits of "indeter-
minate forms," is called "the L'H6pital rule for evaluation of indeterminate
forms." Stories about "evaluation of indeterminate forms" will not injure us
if we resolutely remember that we sometimes find limits but that we never
evaluate 0/0. When we apply the L'Hopital formula (7), we must not fall asleep
at the switch and write the derivative of the quotient f(x)/g(x); we write the
quotient of the derivatives. The following rather simple examples show how
the formula is applied:
(8)hmxs-
limx=2
X--.l x - 1 x-.1 1
limsin x= Iimcos x = 1
x-.o X u-.o 1
lim1 - cos x= limsin x= 0
X--,O X Z- O^1
lim1-cos x= limsin x= limcos x^1
x-.o x2 x-.o 2x x--.o^22
limex-1=limex= 1
x-.o x -.0 1
limex - 1 - x= limex - 1
= limex=^1
x-.o x2 x-.o 2x o T^2
lim 1 1l limx - sin x
x-.o(sinx xl x-.o x sin x
lim 1 - cos x = lim sin x =^0 = 0.x-0xcosx+sinx x-.o2cosx-xsinx 2
14 Show that
lim nx"+1 - (n + 1)x" + I-n(n + 1)
x-.1 (x - 1) 2 2
t Guillaume Francois de 1'Hospital (1661-1704) introduced this formula in a book
.dnalyse des infeniment petits pour intelligence des lignes courbes (Paris, 1696, 182 pp.) which
enjoyed widespread popularity. While there can be objections to tinkering with names of
people, most authorities insist that we must accept evolution from l'Hospital to L'Hopital
quite as cheerfully as we accept evolution from hostel to hotel. Even counterrevolutionists
must recognize that the name is spelled in different ways.