limits - MadAsMathsCreated by T. Madas Created by T. Madas Question 2 (***) Use standard expansions...

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...

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