Limit Excercises Student Version

8/19/2019 Limit Excercises Student Version

1/105

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 ( [email protected] ) 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


2/105

"

#

#$uestion #% lim

# $ $ $

$→+ +

+

"

&

&

"

#

&


3/105

Congratulations YOUR

answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


4/105

%orr&'(t is not #orre#t)

T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


5/105

Substitute x by 1

-a#$ to t"e.uestion

C"e#$ t"e%olution


6/105

"

#

# # # # & lim# # # " $

$ $

$→+ + + += =

+ +


7/105

"

"&

&$uestion "% lim' & $

$ $

$ $→−

− +

"

&

&

"

(

∞


8/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


9/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


10/105

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

(

(


11/105

"

"& &

&

& ) &* lim lim' & ) &*) #*

&lim

) #* "

$ $

$

$ $ $ $

$ $ $ $

$

$

→ →

→

− −=

− + − −

= =

−


12/105

+$uestion &% lim' $ $→∞

+

+

'

(

∞


13/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


14/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


15/105

A number is divided by a

very big value

-a#$ to t"e.uestion

C"e#$ t"e%olution


16/105

hen the denominator is arbitrarily large

then the fraction is very close to zero-

herefore the correct ans.er is zero-


17/105

$uestion '% lim " # $

$→−∞

+

∞

#

−∞

im/ossible


18/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


19/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


20/105

First, find the

domain of f(x)

-a#$ to t"e.uestion

C"e#$ t"e%olution


21/105

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-


22/105

"#

&$uestion 1% lim# $

$

$+→− −

∞

&

"

−∞

&

"

−


23/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


24/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


25/105

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


26/105

"# #

& &

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

# )# *)# * " ( $ $

$ $

$ $ $+ + +→− →−

−= = = −∞

− − + ×


27/105

#

"$uestion 3% lim &# $ $−→

+ ÷−

−∞

&

∞

1


28/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


29/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


30/105

!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


31/105

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# $ $−→

+ = − − ∞ ÷−


32/105

(

" "$uestion 5% lim $

$

$→+ −

∞

" "

(

"

'


33/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


34/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


35/105

!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.


36/105

( (

" " " " " "

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

'" " " " "" " $ $→= = = × =

+ +


37/105

"

"$uestion 6% lim& # $

$

$→−

++ −

#

"

&

∞


38/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


39/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


40/105

!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.


41/105

" "

" " & #

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 "# $

$

→−

+ +

= =


42/105

'

"$uestion +% lim& " # $

$

$→

−− +

&

"

#

'

&

'

(


43/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


44/105

T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


45/105

*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


46/105

' '

" " " & " #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 " " $ $

$ $ $ $

$ $ $ $→ →

− + + − + += =

− − + − +


47/105

&

""

#($uestion #(% lim" 3 $ $ $

$ $→+ −

− −

#&

(

#

5

−

#&

5


48/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


49/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


50/105

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


51/105

& "

"" "

#( ) "*) " 1*lim lim" 3 ) "*)" &* $ $

$ $ $ $ $

$ $ $ $→ →+ − − + +=− − − +

"

"

) " 1* #&

lim )" &* 5 $

$ $

$→

+ +

= =+


52/105

&

"+$uestion ##% lim

& 3 $ $

$→−∞ +

&

(

−∞

+∞


53/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


54/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


55/105

For , al#ays factorie thebiggest $o#er of x from

numerator also for denominator.

-a#$ to t"e.uestion

C"e#$ t"e%olution

∞

∞


56/105

& &

""

" "

+ + +lim lim lim3 3& 3

& & $ $ $

$ $ $

$ $

$ $

→−∞ →−∞ →−∞= =+ + + ÷ ÷

lim & $

$→−∞= = −∞(


57/105

"

"3 &$uestion #"% lim

" # $ $ $

$→∞+ −

−

#

"

"

−∞

+∞


58/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


59/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


60/105

For , al#ays factorie thebiggest $o#er of x from

numerator also for denominator.

-a#$ to t"e.uestion

C"e#$ t"e%olution

∞

∞


61/105

"

""

""

"

3 &#

3 &lim lim#" #

" $ $

$

$ $ $ $

$ $

$

→∞ →∞

+ − ÷+ − =

− − ÷

"

"

3 &#

#lim

# ""

$

$ $

$

→∞

+ − ÷ = =

− ÷

(

((


62/105

"

" "$uestion #&% lim" 1 $

$ $

$→∞+ +

+

"

&

"

(

+∞


63/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


64/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


65/105

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

∞

∞

" $ $=

$ $ $→ −∞ ⇒ = − $ $ $→ ∞ ⇒ =


66/105

""" "" #" "

lim lim" 1 " 1 $ $

$ $ $ $ $

$ $→∞ →∞

+ + ÷+ + =+ +

" " & &lim lim lim1" 1 " 1 "

" $ $ $

$ $ $ $ $

$ $ $

$

→∞ →∞ →∞+ += = = =+ + + ÷

(

(


67/105

"

1 ' & #$uestion #'% lim" $

$ $ $

$→−∞− − +

− +

&−

&

5−

5


68/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


69/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


70/105

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

∞

∞

" $ $=

$ $ $→ −∞ ⇒ = − $ $ $→ ∞ ⇒ =


71/105

""" & #1 '1 ' & #

lim lim" " $ $

$ $ $ $ $ $ $

$ $→−∞ →−∞

− − + ÷− − + =− + − +

(

((

1 ' 1 " 5

lim lim lim" " " $ $ $

$ $ $ $

$ $ $→−∞ →−∞ →−∞

− −

= = =− + − + − +

)%$*

$ $ $→ −∞ ⇒ = −5 5lim lim 5

""#

$ $

$ $

$ $

$

→−∞ →−∞= = = −

− + − + ÷


72/105

" #$uestion #1% lim& " & $

$ $

$ $→−∞ − −+ −

"

&

#

& &+

#

"

un/ene/


73/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


74/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


75/105

om$are the degree of x and .

-a#$ to t"e.uestion

C"e#$ t"e%olution

# $−


76/105

" "

"

"

# #"" #

lim lim& " & " &

& $ $

$ $ $ $ $ $

$ $ $ $

$ $

→−∞ →−∞

− − ÷− − =+ − − − ÷

(

((

(" ( "lim

& ( & $

$

$→−∞

−= =

−


77/105

1 ' & "

' "#

$uestion #3%

#lim

" $

$ $ $ $ $

$ $→−

+ + + + ++ −

#

"

#

"

−

&

"

&

"

−


78/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


79/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


80/105

-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.


81/105

( ) ( ) ( )1 ' & " 1 ' & "

# # $ $ $ $ $ $ $ $ $ $+ + + + + = + + + + +( ) ( ) ( ) ( ) ( )' " ' "# # # # # $ $ $ $ $ $ $ $= + + + + + = + + +

( ) ( ) ( ) ( ) ( )' " ' " " " "

" # # # # # $ $ $ $ $ $ $+ − = − + − = − + + −

( ) ( )( ) ( ) ( ) ( )" " "# # # # # " $ $ $ $ $= − + + = + − +

eside of long division, you may factorie as follo#s&


82/105

( ) ( )( ) ( ) ( )

' "1 ' & "

' " "# #

# ##lim lim

" # # " $ $

$ $ $ $ $ $ $ $

$ $ $ $ $→− →−

+ + +

+ + + + + =

+ − + − +

( )( ) ( )

' "

"#

# & #

lim 3 "# " $

$ $

$ $→−

+ + −

= = =−− +


83/105

$uestion #5%

# &lim

# # $ $ $→ − ÷− −

∞

(

#

#−


84/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


85/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


86/105

-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 ).


87/105

( ) ( )& "

# & # &

# # # # # $ $ $ $ $ $− = −

− − − − + +

( ) ( ) ( ) ( )

"

" "

# & # &

# # # # #

$ $

$ $ $ $ $ $ $

+ + −= − =

− − + + − + +

( ) ( )

"

"

"

# #

$ $

$ $ $

+ −=

− + +


88/105

( ) ( )

"

& "# #

# & "lim lim

# # # # $ $

$ $

$ $ $ $ $→ →

+ − − = ÷− − − + +

( ) ( )( ) ( )

( ) ( )

"

" "# #

# ""lim lim

# # # # $ $

$ $ $ $

$ $ $ $ $ $→ →

− ++ −= =

− + + − + +

( )

( )"#"

lim ## $

$

$ $→

+= = −

− + +


89/105

(

$uestion #6%

sinlim

# tan # tan $

$

$ $→

÷+ − −

∞

(

3

#

"


90/105

e!tQuestion

C"e#$ t"e%olution


91/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


92/105

-a#$ to t"e.uestion

C"e#$ t"e%olution

*ulti$ly denominator by its

con%ugate to find the factor of

sinx .


93/105

(

sin # tan # tanlim

# tan # tan # tan # tan $

$ $ $

$ $ $ $→

+ + −× ÷ ÷

+ − − + + −

( )( )(

sin # tan # tanlim

# tan # tan $

$ $ $

$ $→

+ + − ÷= ÷+ − −

( )(

sin # tan # tanlim

"tan $

$ $ $

$→

+ + −=

( )(cos # tan # tan

lim #" $

$ $ $

→ + + −= =


94/105

( )"$uestion #+%

lim & + # $

$ $ $

→+∞+ − +

∞

(

−∞

3


95/105


answer isCORRECT

e!tQuestion

C"e#$ t"e%olution


96/105


T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


97/105

-a#$ to t"e.uestion

C"e#$ t"e%olution

Factorie the biggest $o#er of x

for radical function first.


98/105

( )" "

"

# #

lim & + # lim & + $ $ $ $ $ $ $ $ $→+∞ →+∞

+ − + = + − + ÷ ÷ ÷

( (

( ) ( )lim & & lim & & $ $

$ $ $ $→+∞ →+∞

= + = + = +∞


99/105

( )"$uestion "(%

lim & + # $

$ $ $

→−∞+ − +

∞

(

−∞

#3


100/105


answer isCORRECT

C"e#$ t"e%olution


101/105

T"in$*ore

C"e#$ t"e%olution

+i,e so*e"ints


102/105

-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.

∞ − ∞

∞

∞


103/105

""

"

& + #lim & + #

& + # $

$ $ $ $ $ $

$ $ $→−∞

− − += + − + × ÷ ÷− − +

(

( )" "

" "

+ + # #lim lim

& + # & + # $ $

$ $ $ $

$ $ $ $ $ $→−∞ →−∞

− − + − ÷= = ÷ ÷− − + − − +


104/105

(

""

#lim

# #& + $

$

$ $

$ $

→−∞

−=

− − + ÷ (

# # #

lim lim & & 3& + $ $

$ $

$ $ $→−∞ →−∞

− −

= = =−− )%$*


105/105

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Limit Excercises Student Version

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