218 Chapter 7Sequences and series
Then becausef(a) 1 = 1 g(a) 1 = 10 ,
and, ifg′(a) is not zero,
(7.29)
This method of finding limits is called l’Hôpital’s rule.
13Ifg′(a) 1 = 10 butf′(a) 1 ≠ 10 the
limit is infinite. Iff′(a)andg′(a) are both zero the process is repeated to give
(7.30)
and so on.
EXAMPLE 7.15Find the limits.
(i).
(ii).
lim()
xa
xe
e
xx→
−
221
=
−!+ !−
=−
!
!
−→− →
xx
x
x
x
353235 1
35
1
6
0
22
as
sinxx( )
x
xx x x
x
−
=
−!+!−−
33532235
lim
sin
x
xx
x
→
−
0
3lim
()
()
()
()
xa
fx
gx
fa
ga
→
=
′′
′′
lim
()
()
()
()
lim
()
xa xa()
fx
gx
fa
ga
fx
→→gx
=
′
′
=
′
′
fx
gx
fa
xa
fa
ga
xa
()
()
()
()
()
()
()
=
′
−
!
′′
′ +
−
!
2
2
ga′′ ′()+
fx
gx
fa x af a
xa
fa
g
()
()
() ( ) ()
()
()
(
=
+−
′−
!
′′
22
aaxaga
xa
)( )() ga
()
+−′ + ()
−
!
′′ +
22
13Guillaume François Antoine de l’Hôpital (1661–1704). French nobleman and amateur mathematician, he
was tutored by Johann Bernoulli in the new calculus. The rule ascribed to him was contained in a letter from
Bernoulli in 1694 and appeared in l’Hôpital’s influential textbook on the calculus, Analyse des infiniment petits
(1696).