L`Hopital`s Rule for Indeterminate Forms - Homework

L’Hopital’s Rule for Indeterminate Forms - Homework
Basic Problems - calculate your answers and check on your calculators
2x2 " x " 3
x !"1
x !1
1.
4 x "1
" "5
lim
x !"1
1
lim
x2 " x " 6
x !3
x"3
2.
2x "1
"5
lim
x !3
1
lim
lim
x2 " x " 3
"3
3. x !3 x " 2
x !1
lim
4.
lim
x !1
x "1
x2 ! 3 " 2
1
2x
2 x2 ! 3
"2
x !1
2x
" lim
2 x2 ! 3
2 " r2 ! 3
5.
"3r 2
" lim 3r r 2 ! 3 " 6
lim
r !1
r !1
2r
"
2 r2 ! 3
x 3 " 3x ! 2
lim 3
x !1 x " x 2 " x ! 1
3x 2 " 3
6. lim
x !1 3 x 2 " 2 x " 1
6x
3
lim
"
x !1 6 x " 2
2
1 " x ! ln x
x 3 " 3x ! 2
7.
"1 x 2 "1
"1 ! 1 x
" lim
"
lim 2
x !1 3 x " 3
x !1 6 x
6
x
1 " ex
8.
1
lim x " "1
x !0 "e
lim
1" r
3
r !1
lim
x !1
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lim
x !0
- 12 -
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x cos x ! sin x
x
WE!!
x !! sin x " $ cos x ! cos x
#0
lim
x (0
1
sin 2 x tan x
3x
-XE!!
sin 2 x sec 2 x $ 2 tan x cos 2 x
lim
#0
x (0
3
sin 2 x + tan x
6x
--E!!
2 cos 2 x $ sec 2 x 1
lim
#
x (0
6
2
tan x ! x
x ! sin x
sec 2 x ! 1
lim
x (0 1 ! cos x
-GE!!
2 sec 2 x tan x
lim
x (0
sin x
2 sec 2 x sec 2 x $ 4 sec 2 x tan 2 x
lim
#2
x (0
cos x
e x ! e! x ! 2 x
x (0
x ! sin x
x
e $ e! x ! 2
lim
x (0 1 ! cos x
-IE!!
e x ! e! x
lim
x (0 sin x
e x $ e! x
#2
lim
x (0 cos x
10 2 x ! 2 x ! 10!2 x
lim
x (0
10 2 x ! 10!2 x
2 ln10 5 10 2 x ! 2 $ 2 ln10 5 10!2 x
-JE!! lim
x (0
2 ln10 5 10 2 x $ 2 ln10 5 10!2 x
4 ln10 ! 2 2 ln10 ! 1
#
4 ln10
2 ln10
lim
x (0
lim
x (0
lim
x (0
lim
x (0
lim
x2 !1
1
#
-KE! lim 2
x (4 4 x $ x
4
2x2 $ 4 x ! 7
#0
-LE!! x (4 x 3 $ 3 x 2 ! 5
lim
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x3
3x 2
6x
6
-TE!! lim x # lim x # lim x # lim x # 0
x (4 e
x (4 e
x (4 e
x (4 e
x2 !1 1
# !!Md%(0>!YiQ%.)(6&!?%*i(!?%1FO
-UE!! lim
x (4 2 x $ 1
2
x2 !1 1
x2
# lim
# !4
-WE!! lim
x (4 2 x $ 1
2 x (!4 x $ 1
1
ln x
# lim x # lim! x # 0
GXE!! lim
x (0 1
x (0 !1
x (0
2
x
x
ex
ex
# lim
# lim xe x # 4
G-E!! lim
x (4 ln x
x (4 1 x
x (4
x2
lim x 2e!3 x # lim 3 x
x (4
x (4 e
GIE!!
2x
2
lim 3 x # lim 3 x # 0
x (4 3e
x (4 9e
lim ! x 2 ln x "
x (0 $
GKE! lim$
x (0
ln x
1x
# lim$
2
3
x (0 !1 x
1x
lim !! x 2 " # 0
lim
x
GGE!! x (4 e
lim
x(
GJE!!
6
2
lim
x(
6
2
x
1
#0
x (4 2 xe x
# lim
cos2 3 x
sec x
lim
#
sec 2 3 x x ( 6 cos x
2
!6 cos 3 x !! sin 3 x "
#0
! sin x
.1 ! cos x 1
lim!csc x ! cot x " # lim0
3
x (0
x (0/ sin x 2
GLE!!
. sin x 1
lim0
3#0
x (0/ cos x 2
x (0 $
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7 1
7 x ! sin x :
1:
lim9
! < # lim9
x (08 sin x
x ; x (08 x sin x <;
7 1 ! cos x :
GTE!! lim9
x (08 x cos x $ sin x <
;
7
:
! sin x
lim9
<#0
x (0 x !! sin x " $ cos x $ cos x
8
;
7 x ! 1 ! ln x :
7 1
1 :
# lim9
lim9
!
<
<
x (18 ln x
x ! 1; x (18 ! x ! 1" ln x ;
:
7
9 1!1 x <
7 x !1 :
< # lim9
9
<
GUE!! lim
x (1 ! x ! 1"
< x (18 x ! 1 $ ln x ;
9
ln
$
x
<;
98 x
7 1 : 1
lim9
<#
x (1 1 $ 1 x
8
; 2
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lim x
!x 2 "
x (0
*y#x
!x 2 "
lim!1 $ x "
* y # !1 $ x "
1
x
ln!1 $ x "
x
IXE!!
ln!1 $ x "
1
# lim
#0
lim
x (4
x
(4
x
1$ x
ln y # 0 * y # e
ln y #
Technically, this is a right - sided limit
sin x
x
x (0
ln y # x 2 ln x
ln x
1x
# lim
#0
GWE!! lim
x (0 1 x 2
x (0 !1 x 3
ln y # 0 * y # 1
.11
lim$ 0 3
x (0 / x 2
1
sin x
.11
* y #0 3
/ x2
.11
ln y # sin x ln0 3 # ! sin x ln x
/ x2
! ln x
!1 x
lim
# lim
I-E!! x (0 $ 1 sin x x (0 $ ! cos x sin 2 x
lim !tan x "
x(6
cos x
* y # !tan x "
cos x
2!
ln y # cos x lntan x
sec 2 x tan x
lntan x
# lim
IGE!! x ( 6 ! 1 cos x x ( 6 ! sin x cos2 x
2
2
lim
sin 2 x
2 sin x cos x
lim$
# lim$
#0
x (0 x cos x
x (0 x !! sin x " $ cos x
cos x
sec 2 x cos2 x
5
# lim
#0
6
6
x( !
sin
tan x x ( 2 ! sin 2 x
2
ln y # 0 * y # 1
ln y # 0 * y # 1
lim
X !
Y):)(+!%3!(80!(9.0!! A!! X A!!X!A!!!g!!A!!!C!!!610!.1)!)*70(01:)*6(0!3%1:+E!!H)*7!(80!3%&&%?)*<!B9!)*+.04()%*>
IIE!! lim$
x (0
x
#0
ln x
IIE!! lim ! !cos x "
x (! 6 "
2
tan x
# 0 !!!!
. 2
1
$ tan x 3 # 4
IJE!! lim ! 0
2
x (! 6 2" / 6 ! 2 x
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