limits - MadAsMathsCreated by T. Madas Created by T. Madas Question 2 (***) Use standard expansions of functions to find the value of the following limit. 0 cos7 1 lim x sin x →

Created by T. Madas

Created by T. Madas

LIMITS

Created by T. Madas

Created by T. Madas

LIMITS BY

STANDARD

EXPANSIONS

Created by T. Madas

Created by T. Madas

Question 1 (***)

a) Write down the first two non zero terms in the expansions of sin3x and cos 2x .

b) Hence find the exact value of

30

3 cos 2 sin3lim

3x

x x x

x→

−

39sin3 32

x x x≈ − , 2cos 2 1 2x x≈ − , 12

−

Created by T. Madas

Created by T. Madas

Question 2 (***)

Use standard expansions of functions to find the value of the following limit.

0

cos7 1lim

sinx

x

x x→

−

.

MM1A , 492

−

Question 3 (***)


5

0

e 5 1lim

sin 4 sin 3

x

x

x

x x→

− −

.

2524

Created by T. Madas

Created by T. Madas

Question 4 (***)

By considering series expansion, determine the value of the following limit.

( )20

2 4lim

ln 1 3x

x x x

x→

− +

−

.

FP3-P , 112

Question 5 (***+)


2

20

cos 3 1limx

x

x→

−

.

FP3-N , 9−

Created by T. Madas

Created by T. Madas

Question 6 (***+)


( )20

ln 1lim cosec

sinx

xx

x→

− +

.

FP3-L , 12

−

Created by T. Madas

Created by T. Madas

Question 7 (****+)


2

0

e 2 4 2lim

x

x

x x

x→

+ + −

.

No credit will be given for using alternative methods such as L’ Hospital’s rule.

MM1G , 52

Created by T. Madas

Created by T. Madas

LIMITS BY

L’HOSPITAL

RULE

Created by T. Madas

Created by T. Madas

Question 1 (**)

Find the value of the following limit

0

coslim

arcsinx

x x

x x→

+

.

FP3-J , 12

Question 2 (**+)


1

lim 2 1x

xx

→∞

−

.

FP3-R , ln 2

Created by T. Madas

Created by T. Madas

Question 3 (***)


2

20

cos 3 1limx

x

x→

−

.

FP3-K , 9−

Question 4 (***)


0

cos7 1lim

sinx

x

x x→

−

.

FP3-Q , 492

−

Created by T. Madas

Created by T. Madas

Question 5 (***)

Use L'Hospital's rule to find the value of the following limit

0

tanlim

sin 2 sinx

x x

x x x→

− − −

.

FP3-O , 27

−

Question 6 (****)


5

0

e 5 1lim

sin 4 sin 3

x

x

x

x x→

− −

.

2524

Created by T. Madas

Created by T. Madas

Question 7 (****)


sin

0lim x

xx

−

→ +

.

MM1B , 1

Question 8 (****+)


( )2

20

sin coslimx

x

x

π

→

.

π

Created by T. Madas

Created by T. Madas

Question 9 (****+)

Find the value of the constant k , given that

( ) ( )2

22

2 2 tan 2lim 5

4 4x

x k x k x

x x→

+ − − − = − +

.

3k =

Question 10 (*****)


( )

2 2

4

sin tanlimx

x x

xπ π→

− −

.

1−

Created by T. Madas

Created by T. Madas

Question 11 (*****)

2 2 2

40

14lim

x

a a x xL

x→

− − − =

, 0a > .

Given that L is finite, determine its value.

164

Created by T. Madas

Created by T. Madas

Question 12 (*****)


3

40 0

1lim sin

y

yx dx

y→

.

14

Created by T. Madas

Created by T. Madas

VARIOUS

LIMITS

Created by T. Madas

Created by T. Madas

Question 1 (**)


2

2

3 7 1lim

5x

x x

x→∞

+ −

+ .

3

Question 2 (**)


3 2

2

2lim

2x

x x x

x→

− − −

− .

7

Created by T. Madas

Created by T. Madas

Question 3 (**+)


3

1 1 1lim

3 3x x x→

− −

.

19

−

Question 4 (**+)

Given that n is a positive integer determine

0

elim

1 e

n x

xx

x

→

− .

0

Created by T. Madas

Created by T. Madas

Question 5 (***)


3

2

8lim

2x

x

x→

−

− .

You may not use the L’ Hospital’s rule in this question.

12

Question 6 (***)

Find the value of the following limit.

lim 5x

x x→∞

+ − .

0

Created by T. Madas

Created by T. Madas

Question 7 (***)

Find the value of the following limit.

32 3lim 1 1x

x x x→∞

+ − +

.

MM1-H , 12

Created by T. Madas

Created by T. Madas

Question 8 (***)

The Fibonacci sequence is given by the recurrence formula

2 1n n nu u u+ += + , 1 1u = , 2 1u = .

It is further given that in this sequence the ratio of consecutive terms converges to a

limit φ , known as the Golden Ratio.

Show, by using the above recurrence formula, that ( )1 1 52

φ = + .

MP2-S , proof

Created by T. Madas

Created by T. Madas

Question 9 (***+)

( )222

n

f n = , n ∈� and ( ) 10001000n

f n = , n ∈� .

Determine whether or not ( )( )

limn

g n

f n→∞

exists.

( )( )

lim 0n

g n

f n→∞

=

Created by T. Madas

Created by T. Madas

Question 10 (***+)

Show clearly without the use of any calculating aid that

6 6 6 6 ... k+ + + + = ,

where k is an integer to be found.

3k =

Question 11 (***+)

2 2 2 2 2 ...x x x x x+ + + + + + + + + + ,

It is given that the above nested radical converges to a limit L , L ∈� .

Determine the range of possible values of x .

SP-T , 94

x ≥ −

Created by T. Madas

Created by T. Madas

Question 12 (***+)

3 3 3 3 34 2 4 2 4 2 4 2 4 ...+ + + + +

Given that the above nested radical converges, determine its limit.

SPX-B , 2L =

Question 13 (****)


2

4

16lim

2x

x

x→

−

− .


32

Created by T. Madas

Created by T. Madas

Question 14 (****)

Find the value of each of the following limits.

a) 1

1lim

1x

x

x→

−

− .

b) ( )

0

sinlim

sinx

kx

x→

.


32

Question 15 (****)


( )0

4 2lim

1x

x

x x→

+ −

+ .


14

Created by T. Madas

Created by T. Madas

Question 16 (****+)


2lim 5x

x x x→∞

+ −

.

MM1-F , 52

Question 17 (****+)


1lim 1

n

n n→∞

+

.

e

Created by T. Madas

Created by T. Madas

Question 18 (****+)

Use two distinct methods to evaluate the following limit

3

28

2lim

9 8x

x

x x→

−

− + .

1

84

Question 19 (****+)


( )( )0

8 cos 1 cos2lim

tan3x

x x

x x→

+ −

.


6

Created by T. Madas

Created by T. Madas

Question 20 (****+)


2 2

21

3 4limx

x x x

x x→

+ + − +

−

.

10

5

Created by T. Madas

Created by T. Madas

Question 21 (****+)


3

8

2lim

8x

x

x→

−

− .


112

Question 22 (****+)

By considering the limit of an appropriate function show that 00 1= .

proof

Created by T. Madas

Created by T. Madas

Question 23 (****+)


2

22

2 3 2lim

4x

x x x

x→

− + − + −

.


12

Question 24 (****+)


2

35

25 5lim

125x

x x

x→

− − − −

.


10 1

60

−

Created by T. Madas

Created by T. Madas

Question 25 (****+)


2lim n n n

xx x x

→∞

− −

, n ∈� .

12

Created by T. Madas

Created by T. Madas

Question 26 (****+)


32

2

1 1lim 1

x

x xx→∞

+ +

.

MM1E , 1

Created by T. Madas

Created by T. Madas

Question 27 (****+)

Use two distinct methods to evaluate the following limit.

1

3 2lim

1x

x x

x→

+ −

− .

MM1E , 32

−

Created by T. Madas

Created by T. Madas

Question 28 (****+)


2lim 3n

n n n→∞

+ −

.


MM1C , 32

Created by T. Madas

Created by T. Madas

Question 29 (****+)

( ) 21f x x= + , x ∈� .

Use the formal definition of the derivative as a limit, to show that

( )21

xf x

x

′ =+

.

proof

Created by T. Madas

Created by T. Madas

Question 30 (****+)

( )2

1

1f x

x=

−, x ∈� , 1x > .


( )( )

322 1

xf x

x

′ = −

−

.

proof

Created by T. Madas

Created by T. Madas

Question 31 (****+)

( ) ( )100

100

100 100

1

1

100r

f x x rx

=

≡ ++ , x ∈� .

Use a formal method to find

( )limx

f x→∞

.

MPX-C , 100

Created by T. Madas

Created by T. Madas

Question 32 (****+)


( )2

2 20

1 coslim

tanx

x

x x→

−

.

SPX-G , 12

Created by T. Madas

Created by T. Madas

Question 33 (*****)

( )1

1

xf x

x

−=

+, x ∈� , 1x < .


( )( ) 2

1

2 1 1f x

x x

′ = −+ −

.

proof

Created by T. Madas

Created by T. Madas

Question 34 (*****)

It is given that for some real constants a and b ,

( )2lim 2 2 2x

x x ax b→+∞

− + − + =

, x ∈� , 0x > .

Determine the value of a and the value of b .

1a = , 3b = −

Created by T. Madas

Created by T. Madas

Question 35 (*****)

Use Leibniz rule and standard series expansions to evaluate the following limit

( )3 40

0

ln 11lim

16

x

x

t tdt

x t→

+

+ .

2

limits - MadAsMathsCreated by T. Madas Created by T. Madas Question 2 (***) Use standard expansions of functions to find the value of the following limit. 0 cos7 1 lim x sin x →

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