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Class 11-commerce RD SHARMA Solutions Maths Chapter 29: Limits

Limits Exercise Ex. 29.1

Solution 1

Solution 2

Solution 3

 

Solution 4

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.

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13(i)

Solution 13(ii)

Solution 13(iii)

Solution 13(iv)

Solution 13(v)

Solution 13(vi)

Solution 13(vii)

Solution 13(viii)

Solution 13(ix)

Solution 13(x)

Solution 13(xi)

Solution 14

Solution 15

Solution 16

Solution 17

 

Solution 18

Solution 19

Solution 20

Solution 21

Solution 6

 

Solution 22

 

Limits Exercise Ex. 29.2

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Limits Exercise Ex. 29.3

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

 

Limits Exercise Ex. 29.4

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 25

 

Limits Exercise Ex. 29.5

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Limits Exercise Ex. 29.6

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

 

Solution 26

  

Limits Exercise Ex. 29.7

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49

Solution 50

Solution 52

Solution 53

Solution 54

Solution 55

Solution 56

Solution 57

Solution 58

Solution 59

Solution 34

 

Solution 51

 

Solution 60

 

Solution 61

 

Solution 62

 

Solution 63

 

Limits Exercise Ex. 29.8

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

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

 

Limits Exercise Ex. 29.9

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Limits Exercise Ex. 29.10

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

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

Solution 3

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

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

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

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

Solution 8

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

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

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

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

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

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

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

Solution 16

Error: the service is unavailable.

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

Solution 18

Error: the service is unavailable.

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

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

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

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

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

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

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

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

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

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

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

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

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

Solution 32

Error: the service is unavailable.

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

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

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

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

Solution 37

Error: the service is unavailable.

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

Solution 39

Error: the service is unavailable.

Solution 40

Solution 41

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

Solution 43

Limits Exercise Ex. 29.11

Solution 1

 

 

Solution 2

 

 

Solution 3

 

 

Solution 4

 

 

Solution 5

 

 

Solution 6

 

Solution 7

 

Solution 8

 

Solution 9

 

Solution 10

 

Limits Exercise Ex. 29VSAQ

Solution 1

limit as x rightwards arrow 0 of fraction numerator square root of 1 minus cos 2 x end root over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator square root of 2 sin squared x end root over denominator x end fraction equals limit as x rightwards arrow 0 of fraction numerator square root of 2 sin x over denominator x end fraction space space o r space limit as x rightwards arrow 0 of fraction numerator minus square root of 2 sin x over denominator x end fraction equals square root of 2 space space space space space space space space o r space space space space space minus square root of 2 text limit   does   not   exists. end text

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

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