McCombs Math 232
Using L'Hospital's Rule
0/0
0/0
()
()
()
()
() () ()
()
# x " tan x & L
# 1 " sec 2 x & L
# "2sec x sec x tan x &
lim %
( = lim %
( = lim %
(
x!0 $ x " sin x ' x!0 $ 1 " cos x ' x!0 $
sin x
'
1.
()
()
#
# sin x & &
% "2sec x sec x %
((
# "2sec x sec x &
$ cos x ' (
%
= lim %
=
lim
( = "2
( x!0 %
x!0 %
sin x
cos
x
$
'
(
%
(
$
'
() ()
()
0/0
2.
3.
( )
() ()
()
( )
" sin 2x % L
" 2cos 2x % 2
lim $
' = lim $
'=
x!0 # 5x & x!0 #
5
& 5
$ 8x9 + 5x 4 + 10x # 7 '
$ 8x9 '
$ 8'
4
!
lim
! lim & ) = #
)
&
)
x! " % #6x9 (
x! " % #6 (
3
15 # 6x9
(
lim
x! " &%
"/"
4.
$ e#x '
$ x 'L
$ 1'
lim & #1 ) = lim & x ) = lim & x ) ! 0
x! " % x ( x! " % e ( x! " % e (
1"
5.
1/ x
lim (1 + x )
x!0+
(
y = 1+ x
Let
1/ x
)
" 1%
! ln y = $ ' ln 1 + x
# x&
( )
(
)
"" 1 %%
$ $# 1 + x '& '
" ln 1 + x % L
"" 1%
%
' =1
lim $ $ ' ln 1 + x ' = lim $
= lim $
'
+
+
+
x
x
1
$
'
#
&
& x!0 #
x!0 #
& x!0
$
'
#
&
0/0
(
Thus,
1/ x
lim (1 + x )
x!0+
(
)
)
= e1 = e
1
McCombs Math 232
Using L'Hospital's Rule
)")
6.
0/0
()
)
# x2
# x 3 " x 2 " 6 ln x &
6 &
lim %
"
( = lim %
(
x " 1' x!1+ $ ln x x " 1 '
x!1+ $ ln x
()
( )(
(
)
( )( )
L
# 3x 2 " 2x " 6 1 / x & "5
= lim %
! ")
(=
x!1+ $ 1 / x x " 1 + ln x 1 ' 0
(
)(
)
7.
##
&
1
0/0
*x *2
%
%
L
#
&
ln 1 + (1 / x)
#
#
$ 1 + (1 / x) ('
1&&
lim x ln % 1 + ( ( = lim %
lim %
( = x!
x!" %$
"%
x ' ' x!" $
$
*x *2
1/ x
'
%
$
")0
(
)
)
&
)(
#
&
1
( = lim
=1
%
( x!" $ 1 + (1 / x) ('
(
'
0/0
#$/$
L
% 1/ x (
% # sin 2 x (
% ln x (
lim tan x ln x = lim '
= lim '
* = lim '
*
x
x!0+
x!0+ & cot x *) x!0+ & # csc 2 x ) x!0+ &
)
((
8.
(
(
0"(#$)
) ( ))
()
()
() ()
L
% #2sin x cos x (
= lim '
* =0
1
x!0+ &
)
0/0
9.
# e x " 1& L
# ex &
lim
= lim
=1
x!0 %$ x (' x!0 %$ 1 ('
10.
# ex " 1 " x & L
# e x " 1& L
# ex & 1
lim %
=
lim
=
lim
( x!0 % 2x ( x!0 % 2 ( = 2
x!0 $
'
$
'
$ '
x2
11.
0/0
0/0
# x
1 2&
e
"
1
"
x
"
x
x
x
x
L
L
L
%
2 ( = lim # e " 1 " x & = lim # e " 1& = lim # e & = 1
lim %
(
%
2 (' x!0 %$ 6x (' x!0 %$ 6 (' 6
x!0 %
x3
( x!0 $ 3x
$
'
0/0
0/0
0/0
2
McCombs Math 232
Using L'Hospital's Rule
0/0
12.
0/0
0/0
# x
# x
1 2 1 3&
1 2&
# ex " 1 " x &
% e " 1" x " 2 x " 6 x ( L
% e " 1" x " 2 x ( L
lim %
( = lim %
( = lim %
3
2 ('
x!0 %
x!0
x4
4x
(
%
( x!0 $ 12x
$
'
$
'
0/0
L
# e x " 1&
# ex & 1
= lim %
=
lim
=
x!0 $ 24x (' x!0 %$ 24 (' 24
13.
$
$ 1 ' '
1 2 1 3
x
n
& e " 1 " x " x " x " # # #&
)x )
2
6
n
"
1
!
%
( ) 1
&
lim &
) = n!
n
x!0 &
x
)
&
)
%
(
(
lim ( sin ( 6x )) =
14.
x!"
15.
lim
x!" %
)
No Limit, since the sine graph oscillates between –1 and 1 as
x ! ".
( )
# sin 6x &
10 ( = 0
$ x
'
( )
sin 6x
1
1
"
"
.
x10
x10
x10
$ 1 '
$ 1 '
Also note that lim & # 10 ) = 0 = lim & 10 ) .
x!" % x (
x!" % x (
Note that
Thus,
( )
!1 " sin 6x " 1 # !
( )
# sin 6x &
10 ( = 0 by the Squeeze Theorem.
$ x
'
lim
x!" %
3