-
LHOPITAL RULELHOPITAL RULE
Function Behavior
Indeterminate Form
LHopital RuleLHopital Rule
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 1
-
Function Behavior Function Behavior
69
1)sin(lim)tan(lim2
02
==
xx
x
xx
xx
pi
How to determines these values by inspecting only from their
equations ?
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 2
56
69lim0lim 2
2
0=
= xx
x
e
x
xxx
-
INDETERMINATE FORMINDETERMINATE FORM
=
=
)(limb)00)(lima)
: FORM ATEINDETERMIN called-so few are There
xf
xf
ax
ax
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 3
=
=
1or or 0)(limd)or 0)(limc)
00xfxf
ax
ax
-
LHOPITAL RULELHOPITAL RULE
DefinitionDefinition
)(')('lim)(
)(limthen
),or or number finite a islimit thisif (i.e., sense infiniteor
finite either thein exists ])('/)('[lim If .0)(lim)(lim that
Suppose
xgxf
xgxf
xgxfxgxf
uxux
uxuxux
=
+
==
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 4
-
LHOPITAL RULELHOPITAL RULE
0/0 Example0/0 Example
1)cos()sin()sin(lim
00)sin(lim
0
===
=x
xx
dxd
x
x
x
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 5
11
)cos()sin(lim0
0
0===
=
=
x
x
x
x
xdxd
dxx
x
-
LHOPITAL RULELHOPITAL RULE
0/0 Example (2)0/0 Example (2)
00
31)cos()sin()sin(lim
00)sin(lim
02
330
30
=
=
=
=
=
xx
x
x
x
xdxd
xxdxd
x
xx
x
xx
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 6
61
6)cos(
6
)sin()sin(lim
00
6)sin(
3
1)cos()sin(lim
0
0
30
0
0
230
0
==
=
==
=
=
=
=
=
=
x
x
x
x
x
x
x
x
xdxd
xdxd
x
xx
x
x
xdxd
xdxd
x
xx
xdx
NOTE:STOP DIFFERENTIATINGWHEN YOU HAVE NOTGOT
INDETERMINATEFORM
-
LHOPITAL RULELHOPITAL RULE
// ExampleExample
01lim
lim
===
=
xx
xdxd
x
e
x
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 7
01lim ====
=
x
x
x
xxx e
edxddx
e
x
-
LHOPITAL LHOPITAL RULERULEOther FormsOther Forms
For limit equation forms beside the previous two (0/0 or /), you
have to change the equation form so their form will produce these
two (0/0 or /) will produce these two (0/0 or /) forms
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 8
-
LHOPITAL RULELHOPITAL RULE
-- Example Example
00
)ln(1)ln(
)ln()1(
1)ln()ln()1(1)ln(lim)ln(
11
lim
)ln(1
1lim
1
11
1
=
+=
+=
+=
=
+
++
+
=
=
x
x
xx
x
xxx
xx
xxdxd
xxxdxd
xx
xxx
xx
x
xx
x
NOW, WE HAVE CHANGEDTHE FORM SO THAT ITS LIMITBECOME ONE OF TWO
(0/0 OR /)
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 9
21
)ln(2)ln(1
)ln()1(
1)ln()ln(
11
lim1
1
1
1
=
+
+=
+=
+
+
+
+
=
=
=
x
x
x
x
x
x
xxdxd
xxxdxd
xx
x
dx
-
LHOPITAL RULELHOPITAL RULE
0 x Example0 x Example
))ln(sin(
)cot())ln(sin(lim))ln(sin()tan(lim
x 0))ln(sin()tan(lim
22
2
==
=
xx
x
xd
x
xxx
xx
pipi
pi
NOW, WE HAVE CHANGEDTHE FORM SO THAT ITS LIMITBECOME ONE OF TWO
(0/0 OR /)
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 10
21
)ln(2)ln(1
)ln()1(
1)ln()ln(
11
lim
0)sin()cos()cot(
))ln(sin())ln(sin()tan(lim
1
1
1
2
2
2
=
+
+=
+=
===
+
+
+
=
=
=
=
x
x
x
x
x
x
x
x
xxdxd
xxxdxd
xx
x
xx
xdxd
xdxd
xx pi
pi
pi
-
LHOPITAL LHOPITAL RULERULEOther Forms (Exponent Forms)Other
Forms (Exponent Forms)
For next exponent forms, we need logarithmic the output (for
example, y) and the input simultaneously to change the origin form
so that its form will be the origin form so that its form will be
like the one of the two (0/0 or /)forms
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 11
-
LHOPITAL RULELHOPITAL RULE
11 ExampleExample
( )( ) ( )
x
xxxyyx
x
x
x
x
+=+==+
=++
)cot(
)cot(0
)tan()1ln(1ln)cot()ln(1
11lim
NOW, WE HAVE CHANGEDTHE FORM SO THAT ITS LIMITBECOME ONE OF TWO
(0/0 OR /)
ADDITIONAL TRICK TO SOLVE EXPONENT FORMOF INDETERMINATE LIMIT
EQUATION
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 12
eeey
xxx
dxd
xdxd
x
xy
x
y
x
x
xx
===
=
+=
+=
+=
+
+
++
=
=
1)ln(
02
0
001)(sec)1(
1
)tan(
)1ln()tan()1ln(lim)ln(lim
)tan(
-
LHOPITAL RULELHOPITAL RULE
00 ExampleExample
( )
( ) )sec())ln(tan()ln()tan(
)tan(lim
)cos(
0)cos(
2
==
=
x
xyyx
x
x
x
xpi
NOW, WE HAVE CHANGEDTHE FORM SO THAT ITS LIMITBECOME ONE OF TWO
(0/0 OR /)
ADDITIONAL TRICK TO SOLVE EXPONENT FORMOF INDETERMINATE LIMIT
EQUATION
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 13
( )
1
0)(sin)cos(
)sec(
))ln(tan()sec())ln(tan(lim)ln(lim
)sec()ln()tan(
0)ln(
2
2
2
22
===
====
==
=
=
eey
x
x
xdxd
xdxd
x
xy
xyyx
y
x
x
xxpi
pi
pipi
-
NONNON--INDETERMINATE FORMINDETERMINATE FORM
Notice the following are not indeterminate forms :
=
=
=
=
)(limb)
00)(lima)
xf
xfax
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 14
==
==
==
=+=
=
=
)(lime)00)(limd)
x )(limd))(limc)
0)(limb)
xfxfxfxfxf
ax
ax
ax
ax
ax
-
( ) )cot( == x
WE DO NOT NEED TO MANIPULATEANY FURTHER BECAUSE THIS IS
NOTINDETERMINATE FORM
NONNON--INDETERMINATE FORMINDETERMINATE FORMExampleExample
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 15
( ) 00)sin(lim )cot(0
==
+
x
xx
-
Any Questions ??Any Questions ??
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 16
-
Resources & Further LearningResources & Further
Learning
Varberg & Purcell, Calculus and Geometry Analytics
9/10/2014 Muhamad Firdaus Syawaludin Lubis, ST., MT. 17
