Request a call back

Join NOW to get access to exclusive study material for best results

Class 12-commerce RD SHARMA Solutions Maths Chapter 9 - Continuity

Continuity Exercise Ex. 9.1

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10(i)


Solution 10(ii)

Solution 10(iii)

Solution 10(iv)

Solution 10(v)

Solution 10(vi)

Solution 10(vii)

Solution 10(viii)

Solution 11

Solution 12

Solution 13

S i n c e space f open parentheses x close parentheses space i s space c o n t i n u o u s space a t space x equals 0 comma space L. H. L i m i t equals R. H. L i m i t.
T h u s comma space w e space h a v e
limit as x rightwards arrow 0 to the power of minus of f open parentheses x close parentheses equals limit as x rightwards arrow 0 to the power of plus of f open parentheses x close parentheses
rightwards double arrow limit as x rightwards arrow 0 to the power of minus of a sin pi over 2 open parentheses x plus 1 close parentheses equals limit as x rightwards arrow 0 to the power of plus of fraction numerator tan x minus sin x over denominator x cubed end fraction
rightwards double arrow a cross times 1 equals limit as x rightwards arrow 0 of fraction numerator tan x minus sin x over denominator x cubed end fraction
rightwards double arrow a equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x over denominator cos x end fraction end style minus sin x over denominator x cubed end fraction
rightwards double arrow a equals limit as x rightwards arrow 0 of fraction numerator fraction numerator sin x over denominator x end fraction open parentheses begin display style fraction numerator 1 over denominator cos x end fraction end style minus 1 close parentheses over denominator x squared end fraction
rightwards double arrow a equals limit as x rightwards arrow 0 of fraction numerator fraction numerator sin x over denominator x end fraction open parentheses begin display style fraction numerator 1 minus cos x over denominator cos x end fraction end style close parentheses over denominator x squared end fraction
rightwards double arrow a equals limit as x rightwards arrow 0 of fraction numerator sin x over denominator x end fraction cross times limit as x rightwards arrow 0 of fraction numerator 1 over denominator cos x end fraction cross times limit as x rightwards arrow 0 of fraction numerator 1 minus cos x over denominator x squared end fraction
rightwards double arrow a equals 1 cross times 1 cross times limit as x rightwards arrow 0 of fraction numerator 1 minus cos x over denominator x squared end fraction
rightwards double arrow a equals limit as x rightwards arrow 0 of fraction numerator 1 minus cos x over denominator x squared end fraction cross times fraction numerator 1 plus cos x over denominator 1 plus cos x end fraction
rightwards double arrow a equals limit as x rightwards arrow 0 of fraction numerator 1 minus cos squared x over denominator x squared open parentheses 1 plus cos x close parentheses end fraction
rightwards double arrow a equals limit as x rightwards arrow 0 of fraction numerator sin squared x over denominator x squared open parentheses 1 plus cos x close parentheses end fraction
rightwards double arrow a equals limit as x rightwards arrow 0 of fraction numerator sin squared x over denominator x squared end fraction cross times limit as x rightwards arrow 0 of fraction numerator 1 over denominator 1 plus cos x end fraction
rightwards double arrow a equals 1 cross times limit as x rightwards arrow 0 of fraction numerator 1 over denominator 1 plus cos x end fraction
rightwards double arrow a equals 1 cross times fraction numerator 1 over denominator 1 plus 1 end fraction
rightwards double arrow a equals 1 half

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Error: the service is unavailable.

 

 

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36(i)

Solution 36(ii)

L e t space x minus 1 equals y
rightwards double arrow x equals y plus 1
T h u s comma space
limit as x rightwards arrow 1 of open parentheses x minus 1 close parentheses tan fraction numerator pi x over denominator 2 end fraction equals limit as y rightwards arrow 0 of y tan fraction numerator pi open parentheses y plus 1 close parentheses over denominator 2 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as y rightwards arrow 0 of y tan open parentheses fraction numerator pi y over denominator 2 end fraction plus pi over 2 close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals minus limit as y rightwards arrow 0 of y c o t fraction numerator pi y over denominator 2 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals minus limit as y rightwards arrow 0 of y fraction numerator cos fraction numerator pi y over denominator 2 end fraction over denominator sin fraction numerator pi y over denominator 2 end fraction end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals minus limit as y rightwards arrow 0 of y fraction numerator cos fraction numerator pi y over denominator 2 end fraction over denominator begin display style fraction numerator open parentheses sin fraction numerator pi y over denominator 2 end fraction close parentheses pi over 2 over denominator pi over 2 end fraction end style end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals minus limit as y rightwards arrow 0 of fraction numerator cos fraction numerator pi y over denominator 2 end fraction over denominator begin display style fraction numerator open parentheses sin fraction numerator pi y over denominator 2 end fraction close parentheses pi over 2 over denominator fraction numerator pi y over denominator 2 end fraction end fraction end style end fraction
space space space space space space space space space space space space space space space space space space space space space space space space equals minus limit as y rightwards arrow 0 of 2 over pi fraction numerator cos fraction numerator pi y over denominator 2 end fraction over denominator begin display style fraction numerator open parentheses sin fraction numerator pi y over denominator 2 end fraction close parentheses over denominator fraction numerator pi y over denominator 2 end fraction end fraction end style end fraction
space space space space space space space space space space space space space space space space space space space space space space space space equals minus 2 over pi limit as y rightwards arrow 0 of cos fraction numerator pi y over denominator 2 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space equals minus 2 over pi
S i n c e space t h e space f u n c t i o n space i s space c o n t i n u o u s comma space L. H. L i m i t equals R. H. L i m i t
T h u s comma space k equals minus 2 over pi

Solution 36(iii)

Solution 36(iv)

Solution 36(v)

Solution 36(vi)

Solution 36(vii)

Solution 36(viii)

Solution 36(ix)

Solution 37

Solution 38

Solution 39(i)

Solution 39(ii)

Solution 40

Solution 41

Solution 42

Given:

At x = 0, we have

  

LHL RHL

So, f(x) is discontinuous at x = 0.

Thus, there is no value of   for which f(x) is continuous at x = 0.

At x = 1, we have

  

LHL = RHL

So, f(x) is continuous at x = 1.

At x = -1, we have

  

LHL = RHL

So, f(x) is continuous at x = -1.

Solution 43

Solution 44

Solution 45

Solution 46

Continuity Exercise Ex. 9.2

Solution 1

Solution 2

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 3(v)

Solution 3(vi)

Solution 3(vii)

Solution 3(viii)

Solution 3(ix)

Solution 3(x)

Solution 3(xi)

Solution 3(xii)

Solution 3(xiii)

Solution 4(i)

Solution 4(ii)

Solution 4(iii)

Solution 4(iv)

Solution 4(v)

Solution 4(vi)

Solution 4(vii)

Solution 4(viii)

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

Continuity Exercise MCQ

Solution 1

Correct option: (c)

Solution 2

Correct option: (a), (b)

Solution 3

Correct option: (a),(d)

Solution 4

Correct option: (c)

Solution 5

Correct option: (d)

Solution 6

Correct option: (c)

Solution 7

Correct option: (c)

Solution 8

Correct option: (b)

Solution 9

Correct option: (b)

Solution 10

Correct option: (d)

Solution 11

Correct option: (b)

Solution 12

Correct option: (a)

Solution 13

Correct option: (d)

Solution 14

Correct option: (c)

Solution 15

Correct option: (c)

Solution 16

Correct option: (c)

Solution 17

Correct option: (c)

Solution 18

Correct option: (c)

Solution 19

Correct option: (d)

Solution 20

Correct option: (b)

Solution 21

Correct option: (a)

Solution 22

Correct option: (a)

Solution 23

Correct option:(d)

Solution 24

Correct option: (c)

Solution 25

Correct option: (b)

Solution 26

Correct option: (b)

Solution 27

Correct option: (a)

Solution 28

Correct option: (a)

Solution 29

Correct option: (b)

Solution 30

Correct option: (a)

Solution 31

Correct option: (a)

Solution 32

Correct option: (d)

Solution 33

Correct option: (b)

Solution 34

Correct option: (b)

Solution 35

Correct option: (a)

Solution 36

Correct option: (d)

Solution 37

Correct option: (c)

Solution 38

Correct option: (b)

Solution 39

Correct option: (b)

Solution 40

Correct option: (b)

Solution 41

Correct option: (d)

Solution 42

Correct option: (a)

Solution 43

We know that, if f(x) and g(x) are continuous then

[f(x) + g(x)], [f(x) - g(x)], f(x)g(x) are continuous functions.

Now,

For f(x) = 0

4x = 0

x = 0

Thus,   is discontinuous at x =0.

Solution 44

The function f(x) = cot x is discontinuous if cot x  

  

Solution 45

Given:

As f(x) is continuous at x = 0, then

  

Hence, the value of f(x) is 0.

Solution 46

Given:

  

As {x} = 0 for integral values of x.

Therefore, domain of f(x) is set of all non-integral values.

Thus, f(x) is discontinuous at all integers.

Solution 47

The function f(x) = [x] is continuous everywhere except for integral values.

Therefore, f(x) = [x] is continuous at x = 1.5.

Solution 48

Given:

As f(x) is continuous at x = 0, we have

  

Continuity Exercise Ex. 9VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Given:

As f(x) is continuous at x = 0, we have

  

Solution 12

Given:

As f(x) is continuous at x = 2, we have

  

Get Latest Study Material for Academic year 24-25 Click here
×