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CHAPTER 8
INDETERMINATE FORMS AND IMPROPER INTEGRALS
8.1 Indeterminate Forms of Type 0/0• Sometimes when a limit is taken, the answer
appears to be 0/0. What does this mean?
• It may mean several things!
• To determine what it really does mean, we use a rule developed by l’Hopital in 1696.
0)(lim)(lim
,)('
)('lim
)(
)(lim
xgxfIF
xg
xf
xg
xf
axax
axax
Common error: Students eagerly apply l’Hopital’s rule to rational functions that
are NOT of the form 0/0!• Find the following limit:
)'''(
0/13
coslim
)''(0/0sin
lim
20
30
sHopitallAPPLYTDONx
x
sHopitallapplyx
x
x
x
Cauchy’s Mean Value Theorem
• Let f & g be differentiable functions on (a,b) and continuous on [a,b]. If g’(x) does not equal 0 for all x in (1,b), then there is a number c in (a,b) such that:
)('
)('
)()(
)()(
cg
cf
agbg
afbf
8.2 Other Indeterminate Forms
• L’Hopital’s rule also holds true for the following indeterminate forms:
)(lim)(lim,)('
)('lim
)(
)(lim xgxfif
xg
xf
xg
xfaxaxaxax
Other indeterminate forms
• Take the logarithm and then apply l’Hopital’s in the following cases:
1,,0 00
Example
1,0ln02cos4
6lim
0/02sin2
3lim
cossin4
3lim
0/0sin2
lim2csc
lim
''1
cotlim
1sincos
lim
''1
)ln(sinlim)ln(sinlim
)ln(sin)ln(sinln)(sin
0)(sinlim
0
0
2
0
2
0
2
3
0
3
2
0
2
0
2
0
00
0
0
eysoyx
xx
x
xx
x
x
x
x
x
sHopitall
x
x
x
xx
sHopitall
x
xxx
xxxyxylet
x
x
xx
xx
xx
xx
xx
x
x
8.3 Improper Integrals: Infinite Limits of Integration
• Improper integrals: If the limits of integration, a and b, are one, the other, or both equal to infinity.
Definition• If the limits on the right exist and have finite
values, then the corresponding improper integrals converge and have those values. Otherwise, the integrals diverge.
b
ab
a
b
aa
b
dxxfdxxf
dxxfdxxf
)(lim)(
)(lim)(
Definition
• If the integral of a function from negative infinity to 0 converges and the integral of the same function from 0 to infinity converses, that the function integrated from negative to positive infinity also converges to the sum of the 2 integrals:
0
0
)()()( dxxfdxxfdxxf
8.4 Improper Integrals: Infinite Integrands
• Let f be continuous on the half-open interval [a,b) and suppose that
• Provided that this limit exists and is finite, in which case we say it converges. Otherwise, it diverges.
t
abt
b
abx
dxxfdxxfthenxf )(lim)(,)(lim
Example
3)1(3lim
)1(lim)1(
03/1
1
0
3/2
1
1
0
3/2
t
t
t
t
x
dxxdxx
Integrands that are infinite at an interior point
• Let f be continuous on [a,b] except at a number c, where a<c<b, and suppose that
• provided both integrals on the right converge. Otherwise, the integral diverges.
b
c
c
a
b
a
cx
dxxfdxxfdxxf
thenxf
)()()(
,)(lim
