Chapter 29 Limits Exercise Ex. 29.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

H e n c e comma space l i m i t space d o e s space n o t space e x i s t.

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13(i)

Solution 13(i)

Question 13(ii)

Solution 13(ii)

Question 13(iii)

Solution 13(iii)

Question 13(iv)

Solution 13(iv)

Question 13(v)

Solution 13(v)

Question 13(vi)

Solution 13(vi)

Question 13(vii)

Solution 13(vii)

Question 13(viii)

Solution 13(viii)

Question 13(ix)

Solution 13(ix)

Question 13(x)

Solution 13(x)

Question 13(xi)

Solution 13(xi)

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 6

Solution 6

Question 22

Solution 22

Chapter 29 Limits Exercise Ex. 29.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 29 Limits Exercise Ex. 29.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Evaluate the following limits:   Solution 34

Chapter 29 Limits Exercise Ex. 29.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 26

Solution 26

Question 27

Evaluate

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 25

Evaluate the following limits:   Solution 25

Chapter 29 Limits Exercise Ex. 29.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Chapter 29 Limits Exercise Ex. 29.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Evaluate the following limits:  Solution 25

Question 26

Evaluate the following limits:  Solution 26

Chapter 29 Limits Exercise Ex. 29.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 34

 Evaluate the following limits:  Solution 34

Question 51

 Evaluate the following limits:  Solution 51

Question 60

Evaluate the following limits:  Solution 60

Question 61

Evaluate the following limits:  Solution 61

Question 62

Evaluate the following limits:  Solution 62

Question 63

Evaluate the following limits:  Solution 63

Chapter 29 Limits Exercise Ex. 29.8

Question 1

limit as x rightwards arrow bevelled pi over 2 of open parentheses pi over 2 minus x close parentheses tan x

Solution 1

limit as x rightwards arrow bevelled pi over 2 of open parentheses pi over 2 minus x close parentheses tan x L e t space y equals pi over 2 minus x a s space x rightwards arrow bevelled fraction numerator pi over denominator 2 comma space space space end fraction space y rightwards arrow 0 limit as x rightwards arrow bevelled pi over 2 of open parentheses pi over 2 minus x close parentheses tan x equals limit as y rightwards arrow 0 of y tan open parentheses pi over 2 minus y close parentheses equals limit as y rightwards arrow 0 of y fraction numerator sin open parentheses pi over 2 minus y close parentheses over denominator cos open parentheses pi over 2 minus y close parentheses end fraction equals limit as y rightwards arrow 0 of y fraction numerator cos y over denominator sin y end fraction equals limit as y rightwards arrow 0 of cos y equals limit as y rightwards arrow 0 of fraction numerator y over denominator sin y end fraction equals 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Evaluate begin mathsize 11px style limit as straight x rightwards arrow straight pi over 8 of fraction numerator cot space 4 straight x space minus space cos space 4 space straight x over denominator left parenthesis straight pi minus 8 straight x right parenthesis cubed end fraction end styleSolution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Evaluate the following limits:  Solution 38

Chapter 29 Limits Exercise Ex. 29.9

Question 1

E v a l u a t e space limit as x rightwards arrow pi of fraction numerator 1 plus cos x over denominator tan squared x end fraction

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Chapter 29 Limits Exercise Ex. 29.10

Question 1

Solution 1

limit as x rightwards arrow 0 of fraction numerator 5 to the power of x minus 1 over denominator square root of 4 plus x end root minus 2 end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses 5 to the power of x minus 1 close parentheses open parentheses square root of 4 plus x end root plus 2 close parentheses over denominator open parentheses square root of 4 plus x end root minus 2 close parentheses open parentheses square root of 4 plus x end root plus 2 close parentheses end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses 5 to the power of x minus 1 close parentheses open parentheses square root of 4 plus x end root plus 2 close parentheses over denominator x end fraction equals 4 space log 5

Question 2

Solution 2

limit as x rightwards arrow 0 of fraction numerator log open parentheses 1 plus x close parentheses over denominator 3 to the power of x minus 1 end fraction equals limit as x rightwards arrow 0 of fraction numerator log open parentheses 1 plus x close parentheses over denominator x end fraction cross times fraction numerator 1 over denominator limit as x rightwards arrow 0 of fraction numerator 3 to the power of x minus 1 over denominator x end fraction end fraction equals fraction numerator 1 over denominator log space 3 end fraction

Question 3

Solution 3

Question 4

Solution 4

limit as x rightwards arrow 0 of fraction numerator a to the power of m x end exponent minus 1 over denominator b to the power of n x end exponent minus 1 end fraction comma space n not equal to 0 equals limit as x rightwards arrow 0 of fraction numerator a to the power of m x end exponent minus 1 over denominator m x end fraction cross times fraction numerator 1 over denominator limit as x rightwards arrow 0 of fraction numerator b to the power of n x end exponent minus 1 over denominator n x end fraction end fraction cross times m over n equals fraction numerator m space log space a over denominator n space log space b end fraction comma space n not equal to 0

Question 5

Solution 5

limit as x rightwards arrow 0 of fraction numerator a to the power of x plus b to the power of x minus 2 over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator a to the power of x minus 1 over denominator x end fraction plus limit as x rightwards arrow 0 of fraction numerator b to the power of x minus 1 over denominator x end fraction equals log space a plus space log space b equals log space left parenthesis a b right parenthesis

Question 6

Solution 6

limit as x rightwards arrow 0 of fraction numerator 9 to the power of x minus 2.6 to the power of x plus 4 to the power of x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses 3 to the power of x close parentheses squared minus 2.3 to the power of x 2 to the power of x plus open parentheses 2 to the power of x close parentheses squared over denominator x squared end fraction equals limit as x rightwards arrow 0 of open parentheses fraction numerator 3 to the power of x minus 2 to the power of x over denominator x end fraction close parentheses squared equals open parentheses limit as x rightwards arrow 0 of fraction numerator 3 to the power of x minus 1 over denominator x end fraction minus limit as x rightwards arrow 0 of open parentheses fraction numerator 2 to the power of x minus 1 over denominator x end fraction close parentheses close parentheses squared equals open parentheses log 3 over 2 close parentheses squared

Question 7

Solution 7

limit as x rightwards arrow 0 of fraction numerator 8 to the power of x minus 4 to the power of x minus 2 to the power of x plus 1 over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses 2 to the power of x minus 1 close parentheses squared open parentheses 2 to the power of x plus 1 close parentheses over denominator x squared end fraction equals limit as x rightwards arrow 0 of open parentheses fraction numerator open parentheses 2 to the power of x minus 1 close parentheses over denominator x end fraction close parentheses squared limit as x rightwards arrow 0 of open parentheses 2 to the power of x plus 1 close parentheses equals 2 space open parentheses log 2 close parentheses squared

Question 8

Solution 8

Question 9

Solution 9

limit as x rightwards arrow 0 of fraction numerator a to the power of x plus b to the power of x plus c to the power of x minus 3 over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator a to the power of x minus 1 over denominator x end fraction plus limit as x rightwards arrow 0 of fraction numerator b to the power of x minus 1 over denominator x end fraction plus limit as x rightwards arrow 0 of fraction numerator c to the power of x minus 1 over denominator x end fraction equals log space a space plus space log space b plus space log space c equals log space left parenthesis a b c right parenthesis

Question 10

Solution 10

L e t space x minus 2 equals h limit as h rightwards arrow 0 of fraction numerator h over denominator log subscript a open parentheses h plus 1 close parentheses end fraction equals limit as h rightwards arrow 0 of fraction numerator log space a over denominator begin display style fraction numerator log open parentheses h plus 1 close parentheses over denominator h end fraction end style end fraction equals log space a

Question 11

Solution 11

limit as x rightwards arrow 0 of fraction numerator 5 to the power of x plus 3 to the power of x plus 2 to the power of x minus 3 over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator 5 to the power of x minus 1 over denominator x end fraction plus limit as x rightwards arrow 0 of fraction numerator 3 to the power of x minus 1 over denominator x end fraction plus limit as x rightwards arrow 0 of fraction numerator 2 to the power of x minus 1 over denominator x end fraction equals log space 5 space plus space log space 3 space plus thin space log space 2 equals log space 30

Question 12

Solution 12

L e t space 1 over x equals h limit as h rightwards arrow 0 of fraction numerator open parentheses a to the power of h minus 1 close parentheses over denominator h end fraction equals log space a

Question 13

Solution 13

limit as x rightwards arrow 0 of fraction numerator a to the power of m x end exponent minus b to the power of n x end exponent over denominator sin space k x end fraction equals limit as x rightwards arrow 0 of fraction numerator a to the power of m x end exponent minus b to the power of n x end exponent over denominator k x space begin display style fraction numerator sin space k x over denominator k x end fraction end style end fraction equals 1 over k limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator open parentheses a to the power of m x end exponent minus b to the power of n x end exponent close parentheses over denominator x end fraction end style over denominator begin display style fraction numerator sin space k x over denominator k x end fraction end style end fraction equals 1 over k log space a to the power of m over b to the power of n

Question 14

Solution 14

limit as x rightwards arrow 0 of fraction numerator a to the power of x plus b to the power of x minus c to the power of x minus d to the power of x over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator a to the power of x minus 1 over denominator x end fraction plus limit as x rightwards arrow 0 of fraction numerator a to the power of x minus 1 over denominator x end fraction minus limit as x rightwards arrow 0 of fraction numerator c to the power of x minus 1 over denominator x end fraction minus limit as x rightwards arrow 0 of fraction numerator d to the power of x minus 1 over denominator x end fraction equals log space a space plus space log space b space minus space log space c space minus space log space d equals log space open parentheses fraction numerator a b over denominator c d end fraction close parentheses

Question 15

Solution 15

limit as x rightwards arrow 0 of fraction numerator e to the power of x minus 1 plus sin space x over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator e to the power of x minus 1 over denominator x end fraction plus limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction equals log space e space plus thin space 1 equals 2

Question 16

Solution 16

Error: the service is unavailable.

Question 17

Solution 17

limit as x rightwards arrow 0 of fraction numerator e to the power of sin space x end exponent minus 1 over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator e to the power of sin space x end exponent minus 1 over denominator sin space x end fraction cross times limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction equals log space e space cross times space 1 equals 1

Question 18

Solution 18

Error: the service is unavailable.

Question 19

Solution 19

limit as x rightwards arrow a of fraction numerator log space x space minus space log space a over denominator x minus a end fraction equals limit as x rightwards arrow a of fraction numerator log begin display style x over a end style over denominator a open parentheses begin display style x over a end style minus 1 close parentheses end fraction l e t space h equals x over a minus 1 equals 1 over a limit as x rightwards arrow a of fraction numerator log begin display style open parentheses h plus 1 close parentheses end style over denominator h end fraction equals 1 over a

Question 20

Solution 20

limit as x rightwards arrow 0 of fraction numerator log open parentheses a plus x close parentheses minus log open parentheses a minus x close parentheses over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator log space open parentheses begin display style fraction numerator a plus x over denominator a minus x end fraction end style close parentheses over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator log space open parentheses begin display style 1 plus fraction numerator 2 x over denominator a minus x end fraction end style close parentheses over denominator fraction numerator 2 x over denominator a minus x end fraction end fraction cross times limit as x rightwards arrow 0 of fraction numerator 2 over denominator a minus x end fraction equals 2 over a

Question 21

Solution 21

limit as x rightwards arrow 0 of fraction numerator log space open parentheses 2 plus x close parentheses plus log open parentheses 0.5 close parentheses over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator log space open parentheses 1 plus begin display style x over 2 end style close parentheses over denominator 2 open parentheses begin display style x over 2 end style close parentheses end fraction equals 1 half

Question 22

Solution 22

limit as x rightwards arrow 0 of fraction numerator log space left parenthesis a plus x right parenthesis minus log space left parenthesis a right parenthesis over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator log open parentheses 1 plus begin display style x over a end style close parentheses over denominator a open parentheses begin display style x over a end style close parentheses end fraction equals 1 over a

Question 23

Solution 23

limit as x rightwards arrow 0 of fraction numerator log open parentheses 3 plus x close parentheses minus log open parentheses 3 minus x close parentheses over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator 3 plus x over denominator 3 minus x end fraction end style close parentheses over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style 1 plus fraction numerator 2 x over denominator 3 minus x end fraction end style close parentheses over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style 1 plus fraction numerator 2 x over denominator 3 minus x end fraction end style close parentheses over denominator fraction numerator 2 x over denominator 3 minus x end fraction end fraction cross times limit as x rightwards arrow 0 of fraction numerator 2 over denominator 3 minus x end fraction equals 2 over 3

Question 24

Solution 24

limit as x rightwards arrow 0 of fraction numerator 8 to the power of x minus 2 to the power of x over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator 8 to the power of x minus 1 over denominator x end fraction minus limit as x rightwards arrow 0 of fraction numerator 2 to the power of x minus 1 over denominator x end fraction equals log space 8 space minus space log space 2 equals log space 4

Question 25

Solution 25

limit as x rightwards arrow 0 of fraction numerator x open parentheses 2 to the power of x minus 1 close parentheses over denominator 1 minus co s space x end fraction equals limit as x rightwards arrow 0 of fraction numerator x open parentheses 2 to the power of x minus 1 close parentheses over denominator 2 sin squared open parentheses begin display style x over 2 end style close parentheses space end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses 2 to the power of x minus 1 close parentheses over denominator x space end fraction cross times limit as x rightwards arrow 0 of fraction numerator x squared over denominator open parentheses begin display style fraction numerator sin open parentheses begin display style x over 2 end style close parentheses space over denominator x over 2 end fraction end style close parentheses squared cross times begin display style x squared over 2 end style end fraction equals 2 log space 2 space equals log space 4 space

Question 26

Solution 26

limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus 1 over denominator log space open parentheses 1 plus x close parentheses end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses square root of 1 plus x end root minus 1 close parentheses open parentheses square root of 1 plus x end root plus 1 close parentheses over denominator log space open parentheses 1 plus x close parentheses open parentheses square root of 1 plus x end root plus 1 close parentheses end fraction equals limit as x rightwards arrow 0 of fraction numerator x over denominator log space open parentheses 1 plus x close parentheses open parentheses square root of 1 plus x end root plus 1 close parentheses end fraction equals limit as x rightwards arrow 0 of fraction numerator 1 over denominator begin display style fraction numerator log space open parentheses 1 plus x close parentheses over denominator x end fraction end style end fraction cross times limit as x rightwards arrow 0 of fraction numerator 1 over denominator open parentheses square root of 1 plus x end root plus 1 close parentheses end fraction equals 1 cross times 1 half equals 1 half

Question 27

Solution 27

limit as x rightwards arrow 0 of fraction numerator log open vertical bar 1 plus x cubed close vertical bar over denominator sin cubed x end fraction equals limit as x rightwards arrow 0 of fraction numerator log open vertical bar 1 plus x cubed close vertical bar over denominator sin cubed x end fraction cross times fraction numerator 1 over denominator limit as x rightwards arrow 0 of open parentheses fraction numerator sin x over denominator x end fraction close parentheses cubed end fraction equals 1 cross times 1 equals 1

Question 28

Solution 28

limit as x rightwards arrow straight pi over 2 of fraction numerator a to the power of c o t space x end exponent minus a to the power of cos space x end exponent over denominator c o t space x space minus space cos space x end fraction equals limit as x rightwards arrow straight pi over 2 of a to the power of cos space x end exponent open square brackets fraction numerator a to the power of c o t space x minus cos x end exponent minus 1 over denominator c o t space x space minus space cos space x end fraction close square brackets equals 1 cross times log space a equals log space a

Question 29

Solution 29

limit as x rightwards arrow 0 of fraction numerator e to the power of x minus 1 over denominator square root of 1 minus cos space x end root end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses e to the power of x minus 1 close parentheses open parentheses square root of 1 plus cos space x end root close parentheses over denominator open parentheses square root of 1 minus cos space x end root close parentheses open parentheses square root of 1 plus cos space x end root close parentheses end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses e to the power of x minus 1 close parentheses open parentheses square root of 1 plus cos space x end root close parentheses over denominator sin space x end fraction B o t h space n u m e r a t o r space a n d space d e n o m i n a t o r space a r e space b o t h space z e r o s space f o r space x equals 0 h e n c e space l i m i t space c a n space n o t space e x i s t

Question 30

Solution 30

limit as x rightwards arrow 0 of fraction numerator e to the power of 5 plus h end exponent minus e to the power of 5 over denominator h end fraction equals e to the power of 5 limit as x rightwards arrow 0 of fraction numerator e to the power of h minus 1 over denominator h end fraction equals e to the power of 5 cross times 1 equals e to the power of 5

Question 31

Solution 31

limit as x rightwards arrow 0 of fraction numerator e to the power of x plus 2 end exponent minus e squared over denominator x end fraction equals e squared limit as x rightwards arrow 0 of fraction numerator e to the power of x minus 1 over denominator x end fraction equals e squared

Question 32

Solution 32

Error: the service is unavailable.

Question 33

Solution 33

limit as x rightwards arrow 0 of fraction numerator e to the power of 3 plus x end exponent minus sin space x minus e cubed over denominator x end fraction equals e cubed limit as x rightwards arrow 0 of fraction numerator e to the power of x minus 1 over denominator x end fraction minus limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction equals e cubed log space e space minus 1 equals e cubed minus 1

Question 34

Solution 34

limit as x rightwards arrow 0 of fraction numerator e to the power of x minus x minus 1 over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator e to the power of x minus 1 over denominator x end fraction minus 1 equals 1 minus 1 equals 0

Question 35

Solution 35

limit as x rightwards arrow 0 of fraction numerator e to the power of 3 x end exponent minus e to the power of 2 x end exponent over denominator x end fraction equals 3 limit as x rightwards arrow 0 of fraction numerator e to the power of 3 x end exponent minus 1 over denominator 3 x end fraction minus limit as x rightwards arrow 0 of fraction numerator e to the power of 2 x end exponent minus 1 over denominator 2 x end fraction equals 3 minus 2 equals 1

Question 36

Solution 36

limit as x rightwards arrow 0 of fraction numerator e to the power of tan space x end exponent minus 1 over denominator tan space x end fraction equals limit as tan space x rightwards arrow 0 of fraction numerator e to the power of tan space x end exponent minus 1 over denominator tan space x end fraction equals 1

Question 37

Solution 37

Error: the service is unavailable.

Question 38

Solution 38

limit as x rightwards arrow 0 of fraction numerator e to the power of tan space x end exponent minus 1 over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator e to the power of tan space x end exponent minus 1 over denominator tan space x end fraction cross times limit as x rightwards arrow 0 of fraction numerator tan space x over denominator x end fraction equals log space e space cross times space 1 equals 1

Question 39

Solution 39

Error: the service is unavailable.

Question 40

Solution 40

Question 41

Evaluate the following limits: limit as x rightwards arrow 0 of fraction numerator a to the power of x minus a to the power of minus x end exponent over denominator x end fractionSolution 41

Question 42

Solution 42

limit as x rightwards arrow 0 of fraction numerator x open parentheses e to the power of x minus 1 close parentheses over denominator 1 minus cos space x end fraction equals limit as x rightwards arrow 0 of fraction numerator x open parentheses e to the power of x minus 1 close parentheses over denominator 2 sin squared open parentheses begin display style x over 2 end style close parentheses end fraction equals limit as x rightwards arrow 0 of fraction numerator open parentheses e to the power of x minus 1 close parentheses over denominator 2 x end fraction cross times limit as x rightwards arrow 0 of 4 over open parentheses begin display style fraction numerator sin open parentheses begin display style x over 2 end style close parentheses over denominator x over 2 end fraction end style close parentheses squared equals 1 half cross times 4 equals 2

Question 43

Evaluate the following limits: limit as x rightwards arrow x over 2 of fraction numerator 2 to the power of minus cos x end exponent minus 1 over denominator open parentheses x minus begin display style pi over 2 end style close parentheses end fractionSolution 43

Chapter 29 Limits Exercise Ex. 29.11

Question 1

Evaluate the following limits: Solution 1

Question 2

Evaluate the following limits: Solution 2

Question 3

Evaluate the following limits: Solution 3

Question 4

Evaluate the following limits: Solution 4

Question 5

Evaluate the following limits: Solution 5

Question 6

Evaluate the following limits: Solution 6

Question 7

Evaluate the following limits: Solution 7

Question 8

Evaluate the following limits: Solution 8

Question 9

Evaluate the following limits: Solution 9

Question 10

Evaluate the following limits: Solution 10

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