Post on 07-Jul-2018
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8/19/2019 Limit Excercises Student Version
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Dear students,
The following questions are designed to review the most important notes that we
need to know to evaluate the limits of the functions.
Every questions includes the hint to solve (in case you cannot find the answer) also
the solution. It is highly recommended to follow the order of the questions (from
the first to the last). esides it is useful to check the solution (there are many
points to learn).
If you did not manage to finish all !" questions at one trial, you may start #y click
on the question and continue from the previous part.
Please do not hesitate to email me ( rezad@utar.edu.my ) your usefulcomments and feedbacks.
All the best,
Mohammad Reza Dehghani
Senior Lecturer
Centre for Foundation studies
University un!u Abdul Rahamn
Level One Leveltwo
Q1 Q6 Q11 Q16
Q2 Q7 Q12 Q17
Q3 Q8 Q13 Q18
Q4 Q9 Q14 Q19
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"
#
#$uestion #% lim
# $ $ $
$→+ +
+
"
&
&
"
#
&
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
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+i,e so*e"ints
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Substitute x by 1
-a#$ to t"e.uestion
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"
#
# # # # & lim# # # " $
$ $
$→+ + + += =
+ +
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"
"&
&$uestion "% lim' & $
$ $
$ $→−
− +
"
&
&
"
(
∞
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Congratulations YOUR
answer isCORRECT
e!tQuestion
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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Since after substitutionx by 3 it is
indeterminate by so try to find a
factor of (x-3) for numerator and
denominator. (factor theorem)
-a#$ to t"e.uestion
C"e#$ t"e%olution
(
(
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"
"& &
&
& ) &* lim lim' & ) &*) #*
&lim
) #* "
$ $
$
$ $ $ $
$ $ $ $
$
$
→ →
→
− −=
− + − −
= =
−
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+$uestion &% lim' $ $→∞
+
+
'
(
∞
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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A number is divided by a
very big value
-a#$ to t"e.uestion
C"e#$ t"e%olution
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hen the denominator is arbitrarily large
then the fraction is very close to zero-
herefore the correct ans.er is zero-
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$uestion '% lim " # $
$→−∞
+
∞
#
−∞
im/ossible
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
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+i,e so*e"ints
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First, find the
domain of f(x)
-a#$ to t"e.uestion
C"e#$ t"e%olution
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Domain of ) * %
#" # (
"
f $
$ $ −
+ ≥ ⇒ ≥
So negative values less than 0(-1
is not /ossible values for f)2*-
herefore, it is im/ossible to
evaluate the limit-
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"#
&$uestion 1% lim# $
$
$+→− −
∞
&
"
−∞
&
"
−
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
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First, factorie denominator.
Second, evaluate each terms
by a value more than -1 but
very close to it.
-a#$ to t"e.uestion
C"e#$ t"e%olution
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"# #
& &
lim lim# )# *)# * $ $
$ $
$ $ $+ +→− →−=− − +
if a//roaches to 0# from right
it means is very close to 0# but it is bigger than 0#-
$
$
So% & ( , # (and # ( but it is very close to zero-
$ $ $< − >
+ >
"# #
& & &lim lim
# )# *)# * " ( $ $
$ $
$ $ $+ + +→− →−
−= = = −∞
− − + ×
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#
"$uestion 3% lim &# $ $−→
+ ÷−
−∞
&
∞
1
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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!hin" about the sign of x-1 #hen x is very close to 1
from left.
-a#$ to t"e.uestion
C"e#$ t"e%olution
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if a//roaches to # from leftit means is very close to # but it is less than #-
$ $
So% # ( but it is very close to zero- $ − <
#
"lim & & very large value 4# $ $−→
+ = − − ∞ ÷−
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(
" "$uestion 5% lim $
$
$→+ −
∞
" "
(
"
'
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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!ry to a$$ly the follo#ing identity(con%ugate)&
-a#$ to t"e.uestion
C"e#$ t"e%olution
" ") *) *a # a # a #− + = −'t hel$s you to find a factor of
ero.
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( (
" " " " " "
lim lim " " $ $
$ $ $
$ $ $→ →
+ − + − + += × + +
( ) ( )( (" "
lim lim" " " " $ $
$ $
$ $ $ $→ →
+ −= =
+ + + +
As you see, factor of ero (x) is fond for numerator and
denominator. And you may cancel of from both.
( )(
# # # " "lim
'" " " " "" " $ $→= = = × =
+ +
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"
"$uestion 6% lim& # $
$
$→−
++ −
#
"
&
∞
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Congratulations YOUR
answer isCORRECT
e!tQuestion
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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!ry to a$$ly the follo#ing identity(con%ugate)&
-a#$ to t"e.uestion
C"e#$ t"e%olution
" ") *) *a # a # a #− + = −'t hel$s you to find a factor of
ero.
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" "
" " & #
lim lim& # & # & # $ $
$ $ $
$ $ $→− →−
+ + + +
= ×+ − + − + +
( ) ( ) ( ) ( )" "
" " " " " "lim lim
& # " $ $
$ $ $ $
$ $→− →−
+ + + + + += =
+ − +As you see, factor of ero (x) is found for numerator
and denominator. And you may cancel off from both.
( )"
& #
lim "# $
$
→−
+ +
= =
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'
"$uestion +% lim& " # $
$
$→
−− +
&
"
#
'
&
'
(
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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*ulti$ly con%ugate of numeratorand denominator to fractional
function to find the factor of
ero.
-a#$ to t"e.uestion
C"e#$ t"e%olution
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' '
" " " & " #lim lim
& " # & " # " & " # $ $
$ $ $ $
$ $ $ $→ →
− − + + += × ×
− + − + + + +
As you see, factor of ero (+-x) is found for numerator
and denominator. And you may cancel off from both.
( ) ( )( ) ( )
( )( )' '
' & " # & " # &lim lim'" ' " " " $ $
$ $ $
$ $ $→ →
− + + + += = =− + +
( ) ( )
( ) ( )
( ) ( )
( ) ( )' '
' & " # ' & " #lim lim
+ " # " 6 " " $ $
$ $ $ $
$ $ $ $→ →
− + + − + += =
− − + − +
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&
""
#($uestion #(% lim" 3 $ $ $
$ $→+ −
− −
#&
(
#
5
−
#&
5
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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y doing the long division (orfactoriing) find a factor of (x-)
for numerator and denominator.
-a#$ to t"e.uestion
C"e#$ t"e%olution
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& "
"" "
#( ) "*) " 1*lim lim" 3 ) "*)" &* $ $
$ $ $ $ $
$ $ $ $→ →+ − − + +=− − − +
"
"
) " 1* #&
lim )" &* 5 $
$ $
$→
+ +
= =+
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&
"+$uestion ##% lim
& 3 $ $
$→−∞ +
&
(
−∞
+∞
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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For , al#ays factorie thebiggest $o#er of x from
numerator also for denominator.
-a#$ to t"e.uestion
C"e#$ t"e%olution
∞
∞
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& &
""
" "
+ + +lim lim lim3 3& 3
& & $ $ $
$ $ $
$ $
$ $
→−∞ →−∞ →−∞= =+ + + ÷ ÷
lim & $
$→−∞= = −∞(
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"
"3 &$uestion #"% lim
" # $ $ $
$→∞+ −
−
#
"
"
−∞
+∞
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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For , al#ays factorie thebiggest $o#er of x from
numerator also for denominator.
-a#$ to t"e.uestion
C"e#$ t"e%olution
∞
∞
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"
""
""
"
3 &#
3 &lim lim#" #
" $ $
$
$ $ $ $
$ $
$
→∞ →∞
+ − ÷+ − =
− − ÷
"
"
3 &#
#lim
# ""
$
$ $
$
→∞
+ − ÷ = =
− ÷
(
((
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"
" "$uestion #&% lim" 1 $
$ $
$→∞+ +
+
"
&
"
(
+∞
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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For , al#ays factorie the biggest $o#er of x
from numerator also for denominator. ut #henthere is suare root, factorie the biggest
$o#er of x inside the radical first.
areful about&
-a#$ to t"e.uestion
C"e#$ t"e%olution
∞
∞
" $ $=
$ $ $→ −∞ ⇒ = − $ $ $→ ∞ ⇒ =
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""" "" #" "
lim lim" 1 " 1 $ $
$ $ $ $ $
$ $→∞ →∞
+ + ÷+ + =+ +
" " & &lim lim lim1" 1 " 1 "
" $ $ $
$ $ $ $ $
$ $ $
$
→∞ →∞ →∞+ += = = =+ + + ÷
(
(
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"
1 ' & #$uestion #'% lim" $
$ $ $
$→−∞− − +
− +
&−
&
5−
5
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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For , al#ays factorie the biggest $o#er of x
from numerator also for denominator. ut #henthere is suare root, factorie the biggest
$o#er of x inside the radical first.
areful about&
-a#$ to t"e.uestion
C"e#$ t"e%olution
∞
∞
" $ $=
$ $ $→ −∞ ⇒ = − $ $ $→ ∞ ⇒ =
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""" & #1 '1 ' & #
lim lim" " $ $
$ $ $ $ $ $ $
$ $→−∞ →−∞
− − + ÷− − + =− + − +
(
((
1 ' 1 " 5
lim lim lim" " " $ $ $
$ $ $ $
$ $ $→−∞ →−∞ →−∞
− −
= = =− + − + − +
)%$*
$ $ $→ −∞ ⇒ = −5 5lim lim 5
""#
$ $
$ $
$ $
$
→−∞ →−∞= = = −
− + − + ÷
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" #$uestion #1% lim& " & $
$ $
$ $→−∞ − −+ −
"
&
#
& &+
#
"
un/ene/
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
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om$are the degree of x and .
-a#$ to t"e.uestion
C"e#$ t"e%olution
# $−
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" "
"
"
# #"" #
lim lim& " & " &
& $ $
$ $ $ $ $ $
$ $ $ $
$ $
→−∞ →−∞
− − ÷− − =+ − − − ÷
(
((
(" ( "lim
& ( & $
$
$→−∞
−= =
−
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1 ' & "
' "#
$uestion #3%
#lim
" $
$ $ $ $ $
$ $→−
+ + + + ++ −
#
"
#
"
−
&
"
&
"
−
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Congratulations YOUR
answer isCORRECT
e!tQuestion
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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-a#$ to t"e.uestion
C"e#$ t"e%olution
y doing the long division (or
factoriing) find a factor of (x1)
for numerator and denominator.
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( ) ( ) ( )1 ' & " 1 ' & "
# # $ $ $ $ $ $ $ $ $ $+ + + + + = + + + + +( ) ( ) ( ) ( ) ( )' " ' "# # # # # $ $ $ $ $ $ $ $= + + + + + = + + +
( ) ( ) ( ) ( ) ( )' " ' " " " "
" # # # # # $ $ $ $ $ $ $+ − = − + − = − + + −
( ) ( )( ) ( ) ( ) ( )" " "# # # # # " $ $ $ $ $= − + + = + − +
eside of long division, you may factorie as follo#s&
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( ) ( )( ) ( ) ( )
' "1 ' & "
' " "# #
# ##lim lim
" # # " $ $
$ $ $ $ $ $ $ $
$ $ $ $ $→− →−
+ + +
+ + + + + =
+ − + − +
( )( ) ( )
' "
"#
# & #
lim 3 "# " $
$ $
$ $→−
+ + −
= = =−− +
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$uestion #5%
# &lim
# # $ $ $→ − ÷− −
∞
(
#
#−
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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-a#$ to t"e.uestion
C"e#$ t"e%olution
Sim$lify first by ma"e them same
denominator.
Find the factor of (x-1 ).
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( ) ( )& "
# & # &
# # # # # $ $ $ $ $ $− = −
− − − − + +
( ) ( ) ( ) ( )
"
" "
# & # &
# # # # #
$ $
$ $ $ $ $ $ $
+ + −= − =
− − + + − + +
( ) ( )
"
"
"
# #
$ $
$ $ $
+ −=
− + +
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( ) ( )
"
& "# #
# & "lim lim
# # # # $ $
$ $
$ $ $ $ $→ →
+ − − = ÷− − − + +
( ) ( )( ) ( )
( ) ( )
"
" "# #
# ""lim lim
# # # # $ $
$ $ $ $
$ $ $ $ $ $→ →
− ++ −= =
− + + − + +
( )
( )"#"
lim ## $
$
$ $→
+= = −
− + +
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(
$uestion #6%
sinlim
# tan # tan $
$
$ $→
÷+ − −
∞
(
3
#
"
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e!tQuestion
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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-a#$ to t"e.uestion
C"e#$ t"e%olution
*ulti$ly denominator by its
con%ugate to find the factor of
sinx .
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(
sin # tan # tanlim
# tan # tan # tan # tan $
$ $ $
$ $ $ $→
+ + −× ÷ ÷
+ − − + + −
( )( )(
sin # tan # tanlim
# tan # tan $
$ $ $
$ $→
+ + − ÷= ÷+ − −
( )(
sin # tan # tanlim
"tan $
$ $ $
$→
+ + −=
( )(cos # tan # tan
lim #" $
$ $ $
→ + + −= =
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( )"$uestion #+%
lim & + # $
$ $ $
→+∞+ − +
∞
(
−∞
3
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Congratulations YOUR
answer isCORRECT
e!tQuestion
C"e#$ t"e%olution
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%orr&'(t is not #orre#t)
T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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-a#$ to t"e.uestion
C"e#$ t"e%olution
Factorie the biggest $o#er of x
for radical function first.
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( )" "
"
# #
lim & + # lim & + $ $ $ $ $ $ $ $ $→+∞ →+∞
+ − + = + − + ÷ ÷ ÷
( (
( ) ( )lim & & lim & & $ $
$ $ $ $→+∞ →+∞
= + = + = +∞
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( )"$uestion "(%
lim & + # $
$ $ $
→−∞+ − +
∞
(
−∞
#3
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Congratulations YOUR
answer isCORRECT
C"e#$ t"e%olution
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T"in$*ore
C"e#$ t"e%olution
+i,e so*e"ints
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-a#$ to t"e.uestion
C"e#$ t"e%olution
!he limit is li"e #hich is
indeterminate limit. So first multi$ly by
its con%ugate to change it as a then
factorie the biggest $o#er of x.
∞ − ∞
∞
∞
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""
"
& + #lim & + #
& + # $
$ $ $ $ $ $
$ $ $→−∞
− − += + − + × ÷ ÷− − +
(
( )" "
" "
+ + # #lim lim
& + # & + # $ $
$ $ $ $
$ $ $ $ $ $→−∞ →−∞
− − + − ÷= = ÷ ÷− − + − − +
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(
""
#lim
# #& + $
$
$ $
$ $
→−∞
−=
− − + ÷ (
# # #
lim lim & & 3& + $ $
$ $
$ $ $→−∞ →−∞
− −
= = =−− )%$*
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E OE'