RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity

The solutions of this chapter are designed by our experts to boost confidence among students in understanding the concepts covered in this chapter and methods to solve problems in a shorter period. RD Sharma Solutions, Maths Chapter 9, for Class 12, help students who aspire to obtain a good academic score in the exam. This chapter of RD Sharma Solutions for Class 12 mainly focuses on the concept of continuity. To know more about this topic students can download the RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity from the below provided links. This chapter explains continuity and its applications. Let us have a look at some of the important concepts that are discussed in this chapter.

  • The intuitive notion of continuity
  • Continuity at a point
  • Algebra of continuous function
    • Testing the continuity of a function at a point when the function has the same definition on both sides of the given point
    • Testing continuity of a function at a point when the function has different definitions on both sides of the given point
    • Finding the values of a constant given in the definition of a function when it is continuous at an indicated point
    • On continuity of composite function
  • Continuity on an interval
  • Continuity on an open interval
  • Continuity on a closed interval
  • Definition and meaning of continuous function
  • Definition and meaning of everywhere continuous function
  • Properties of continuous function
    • Testing the continuity of a function in its domain
    • Finding the values of a constant given in the definition of a function when it is continuous on its domain

RD Sharma Solutions For Class 12 Maths Chapter 9 Continuity:-Download PDF Here

RD Sharma Class 12 Maths Solutions Chapter 9 Continuity
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 1
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 2
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 3
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 4
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 5
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 6
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 7
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 8
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 9
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 10
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 11
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 12
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 13
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 14
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 15
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 16
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 17
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 18
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 19
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 20
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 21
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 22
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 23
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 24
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 25
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 26
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 27
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 28
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 29
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 30
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 31
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 32
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 33
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 34
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 35
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 36
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 37
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 38
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 39
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 40
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 41
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 42
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 43
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 44
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 45
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 46
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 47
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 48
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 49
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 50
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 51
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 52
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 53
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 54
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 55
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 56
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 57
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 58
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 59
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 60
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 61
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 62
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 63
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 64
RD Sharma Class 12 Maths Solutions Chapter 9 Continuity 65

Access answers to Maths RD Sharma Solutions For Class 12 Chapter 9 – Continuity

Exercise 9.1 Page No: 9.16

1. Test the continuity of the following function at the origin:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 1

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 2

Consider LHL at x = 0

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 3

2. A function f(x) is defined as

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 4

Show that f(x) is continuous at x = 3.

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 5

3. A function f(x) is defined as

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 6

Show that f(x) is continuous at x = 3.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 7
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 8

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 9

Find whether f(x) is continuous at x = 1.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 10
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 11
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 12

Find whether f(x) is continuous at x = 0.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 13
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 14
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 15

Find whether f(x) is continuous at x = 0.

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 16
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 17
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 18
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 19

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 20
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 21
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 22
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 23

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 24
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 25
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 26

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 27
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 28

10. Discuss the continuity of the following functions at the indicated point(s):

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 29

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 30
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 31
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 32

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 33
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 34

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 35
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 36
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 37

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 38
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 39
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 40

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 41
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 42
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 43

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 44
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 45
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 46

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 47
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 48
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 49

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 50
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 51

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 52
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 53

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 54
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 55
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 56

13. Find the value of a for which the function f defined by

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 57

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 58
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 59
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 60

14. Examine the continuity of the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 61

Also sketch the graph of this function.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 62
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 63

Hence f (x) is discontinuous at x = 0

15. Discuss the continuity of the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 64

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 65

16. Discuss the continuity of the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 66

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 67
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 68

17. Discuss the continuity of

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 69

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 70
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 71

18. For what value of k is the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 72

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 73

19. Determine the value of the constant k so that the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 74

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 75

20. For what value of k is the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 76

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 77
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 78

21. Determine the value of the constant k so that the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 79

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 80
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 81

22. Determine the value of the constant k so that the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 82

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 83

23. Find the values of a so that the function

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 84

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 85

Exercise 9.2 Page No: 9.34

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 86

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 87 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 88

To prove it everywhere continuous we need to show that at every point in the domain of f(x) [domain is nothing but a set of real numbers for which function is defined]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 89

Clearly from definition of f(x), f(x) is defined for all real numbers.

Now we need to check continuity for all real numbers.

Let c is any random number such that c < 0 [thus c being a random number, it can include all negative numbers]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 90

We can say that f(x) is continuous for all x < 0

Now, let m be any random number from the domain of f such that m > 0

Thus m being a random number, it can include all positive numbers]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 91

Therefore we can say that f(x) is continuous for all x > 0

As zero is a point at which function is changing its nature so we need to check LHL, RHL separately

f (0) = 0 + 1 = 1 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 92

Thus LHL = RHL = f (0).

Therefore f (x) is continuous at x = 0

Hence, we proved that f is continuous for x < 0; x > 0 and x = 0

Thus f(x) is continuous everywhere.

Hence, proved.

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 93

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 94 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 95

Here we have,

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 96 …….equation 1

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Function is changing its nature (or expression) at x = 0, so we need to check its continuity at x = 0 first.

We know that from the definition of mod function we have

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 97

f (0) = 0 [using equation 1]

Clearly, LHL ≠ RHL ≠ f (0)

∴ Function is discontinuous at x = 0

Let c be any real number such that c > 0

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 98

Therefore f (x) is continuous everywhere for x > 0.

Let c be any real number such that c < 0

Therefore f (c) = 
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 99

[Using equation 1 and idea of mod function]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 100

Therefore f (x) is continuous everywhere for x < 0.

Hence, we can conclude by stating that f (x) is continuous for all Real numbers except zero that is discontinuous at x = 0.

3. Find the points of discontinuity, if any, of the following functions:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 101

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 102 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 103

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Function is changing its nature (or expression) at x = 1, so we need to check its continuity at x = 1.

Clearly, f (1) = 4 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 104

∴ f (x) is discontinuous at x = 1.

Let c be any real number such that c ≠ 0

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 105
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 106

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 107 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 108

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Function is changing its nature (or expression) at x = 2, so we need to check its continuity at x = 2 first.

Clearly, f (2) = 16 [from equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 109

∴ f (x) is continuous at x = 2.

Let c be any real number such that c ≠ 0

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 110
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 111

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 112 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 113

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c < 0 [thus c being a random number, it can include all negative numbers]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 114

We can say that f(x) is continuous for all x < 0

Now, let m be any random number from the domain of f such that m > 0

Thus m being a random number, it can include all positive numbers]

f (m) = 2m + 3 [from equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 115

We can say that f(x) is continuous for all x > 0

As zero is a point at which function is changing its nature so we need to check LHL, RHL separately

f (0) = 2 × 0 + 3 = 3 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 116
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 117

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 118 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 119

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c ≠ 0 [thus c being a random number, it can include all numbers except 0]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 120

We can say that f(x) is continuous for all x ≠ 0

As zero is a point at which function is changing its nature, so we need to check the continuity here.

f (0) = 4 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 121
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 122

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 123 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 124
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 125

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c ≠ 0 [thus c being a random number, it can include all numbers except 0]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 126

We can say that f(x) is continuous for all x ≠ 0

As zero is a point at which function is changing its nature, so we need to check the continuity here.

f (0) = 5 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 127
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 128

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

https://gradeup-question-images.grdp.co/liveData/PROJ23776/1543472441418396.png 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 130

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c ≠ 0 [thus c being a random number, it can include all numbers except 0]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 131

We can say that f(x) is continuous for all x ≠ 0

As zero is a point at which function is changing its nature so we need to check the continuity here.

f (0) = 10 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 132
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 133

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 134 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 135

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c ≠ 0 [thus c being random number, it is able to include all numbers except 0]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 136

We can say that f(x) is continuous for all x ≠ 0

As x = 0 is a point at which function is changing its nature so we need to check the continuity here.

Since, f (0) = 7 [from equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 137

Log (1+x) and ex in its Taylor form.

From sandwich theorem numerator and denominator conditions also hold for this limit

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 138
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 139

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 140 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 141
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 142

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c < 1 [thus c being a random number, it can include all numbers less than 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 143

We can say that f(x) is continuous for all x < 1

As x = 1 is a point at which function is changing its nature, so we need to check the continuity here.

f (1) = | 1 – 3 | = 2 [from equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 144

Thus LHL = RHL = f (1)

∴ f (x) is continuous at x = 1

Now, again f(x) is changing its nature at x = 3, so we need to check continuity at x = 3

f (3) = 3 – 3 = 0 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 145

Thus LHL = RHL = f (3)

∴ f (x) is continuous at x = 3

For x > 3; f(x) = x–3 whose plot is linear, so it is continuous for all x > 3

Similarly, for 1 < x < 3, f(x) = 3 – x whose plot is again a straight line and thus continuous for all point in this range.

Hence, f(x) is continuous for all real x.

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 146

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 147

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 148

Similarly, we can define it for variable x, if x ≥ 0 |x| = x

If x < 0 |x| = (–x)

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 149
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 150

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c < –3 [thus c being random number, it is able to include all numbers less than –3]

f (c) = 3 – c [from equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 151

Thus LHL = RHL = f (–3)

∴ f (x) is continuous at x = –3

Let c is any random number such that –3 < m < 3 [thus c being random number, it is able to include all numbers between –3 and 3]

f (c) = -2m [ using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 152

We can say that f(x) is continuous for all –3 < x < 3

Now, again f(x) is changing its nature at x = 3, so we need to check continuity at x = 3

f (3) = 6 × 3 + 2 = 20 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 153

Thus LHL ≠ RHL

∴ f (x) is discontinuous at x = 3

For x > 3; f(x) = 6x + 2 whose plot is linear, so it is continuous for all x > 3

Hence, f(x) is continuous for all real x except x = 3

There is only one point of discontinuity at x = 3

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 154

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 155

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 156

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c < 1 [thus c being random number, it is able to include all numbers less than 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 157

We can say that f(x) is continuous for all x < 1

As x = 1 is a point at which function is changing its nature so we need to check the continuity here.

f (1) = 110 = 1 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 158

Hence, f(x) is continuous for all real x

There no point of discontinuity. It is everywhere continuous

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 159

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 160 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 161

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c < 0 [thus c being a random number, it can include all numbers less than 0]

f (c) = 2c

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 162

We can say that f(x) is continuous for all x < 0

As x = 0 is a point at which function is changing its nature, so we need to check the continuity here.

f (0) = 0 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 163

Thus LHL = RHL = f (0)

∴ f (x) is continuous at x = 0

Let m is any random number such that 0 < m < 1 [thus m being a random number, it can include all numbers greater than 0 and less than 1]

f (m) = 0 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 164

We can say that f(x) is continuous for all 0 < x < 1

As x = 1 is again a point at which function is changing its nature, so we need to check the continuity here.

f (1) = 0

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 165

∴ f (x) is discontinuous at x = 1

Let k is any random number such that k > 1 [thus k being a random number, it can include all numbers greater than 1]

f (k) = 4k [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 166

We can say that f(x) is continuous for all x > 1

Hence, f(x) is continuous for all real value of x, except x =1

There is a single point of discontinuity at x = 1

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 167

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 168 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 19

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

Let c is any random number such that c ≠ 0 [thus c being a random number, it can include all numbers except 0]

f (c) = sin c – cos c [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 170

We can say that f(x) is continuous for all x ≠ 0

As zero is a point at which function is changing its nature, so we need to check the continuity here.

f (0) = –1 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 171

∴ f (x) is continuous at x = 0

Hence, f is continuous for all x.

f (x) is continuous everywhere.

No point of discontinuity.

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 172

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 173 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 174

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

For x < –1, f(x) is having a constant value, so the curve is going to be straight line parallel to x–axis.

So, it is everywhere continuous for x < –1.

Similarly for –1 < x < 1, plot on X–Y plane is a straight line passing through origin.

So, it is everywhere continuous for –1 < x < 1.

And similarly for x > 1, plot is going to be again a straight line parallel to x–axis

∴ it is also everywhere continuous for x > 1

As x = –1 is a point at which function is changing its nature so we need to check the continuity here.

f (–1) = –2

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 175

∴ f (x) is continuous at x = –1

Also at x = 1 function is changing its nature so we need to check the continuity here too.

f (1) = 2 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 176

∴ f (x) is continuous at x = 1

Thus, f(x) is continuous everywhere and there is no point of discontinuity.

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 177

4. In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous:

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 178

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 179

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 180

Function is defined for all real numbers and we need to find the value of k so that it is continuous everywhere in its domain (domain = set of numbers for which f is defined)

As, for x ≠ 0 it is just a combination of trigonometric and linear polynomial both of which are continuous everywhere.

As x = 0 is only point at which function is changing its nature so it needs to be continuous here.

f (0) = 3k [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 181
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 182

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 183 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 184

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere.

From equation 1, it is clear that f(x) is changing its expression at x = 2

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 185
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 186

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 187 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 188

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = 0

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 189

As above equality never holds true for any value of k

k = not defined

No such value of k is possible for which f(x) is continuous everywhere.

f (x) will always have a discontinuity at x = 0

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 190

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

https://gradeup-question-images.grdp.co/liveData/PROJ23776/1543472540835883.png 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 192

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = 3

Given, f(x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 193

∴ 3a + b = 2 ……………….Equation 2

Also from equation 1, it is clear that f(x) is also changing its expression at x = 5

Given, f(x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 194
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 195

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 196 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 197

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = –1

Given, f(x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 198

∴ a + b = 4 ……………….Equation 2

Also from equation 1, it is clear that f(x) is also changing its expression at x = 0

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 199
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 200

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 201 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 202

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = 0

Given, f(x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 203
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 204

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 205 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 206

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = 2

Given, f(x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 207

∴ 2a + b = 5 ……………….Equation 2

Also from equation 1, it is clear that f(x) is also changing its expression at x = 10

Given, f(x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 208

∴ 10a + b = 21 ……………….Equation 3

As, b = 21 – 10a

Putting value of b in equation 2, we get

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 209
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 210

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 211 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 212

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = π/2

Given, f(x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 213
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 214

Is continuous on [0, ∞). Find the most suitable values of a and b.

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

https://gradeup-question-images.grdp.co/liveData/PROJ23776/1543472595240417.png 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 216
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 217

The function is defined for [0, ∞) and we need to find the value of a and b so that it is continuous everywhere in its domain (domain = set of numbers for which f is defined)

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = 1

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 218

Also from equation 1, it is clear that f(x) is also changing its expression at x = √2

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 219

∴ b2 – 2b = a ……………….Equation 3

From equation 2, a = –1

b2 – 2b = –1

⇒ b2 – 2b + 1 = 0

⇒ (b – 1)2 = 0

∴ b = 1 when a = –1

Putting a = 1 in equation 3:

b2 – 2b = 1

⇒ b2 – 2b – 1 = 0

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 220

Thus,

For a = –1; b = 1

For a = 1; b = 1 ± √2

6. Find the values of a and b so that the function f (x) defined by

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 221

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 222 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 223

Function is defined for [0, π] and we need to find the value of a and b so that it is continuous everywhere in its domain (domain = set of numbers for which f is defined)

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = π/4

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 224

∴ a + b = – π/4 …………… equation 2

Also from equation 1, it is clear that f(x) is also changing its expression at x = π/2

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 225

b = – a – b

∴ a = –2b ……………….Equation 3

Putting value of a from equation 3 to equation 2

∴ –2b + b = – π/4

⇒ b = π/4

∴ a = –2 × (π/4)

= –π/2

Thus, a = –π/2 and b = π/4

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 226

If f is continuous on [0, 8], find the values of a and b.

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 227 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 228

Function is defined for [0, 8] and we need to find the value of a and b so that it is continuous everywhere in its domain (domain = set of numbers for which f is defined)

To find the value of constants always try to check continuity at the values of x for which f(x) is changing its expression.

As most of the time discontinuities are here only, if we make the function continuous here, it will automatically become continuous everywhere

From equation 1, it is clear that f(x) is changing its expression at x = 2

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 229

4+2a + b = 8

∴ 2a + b = 4

∴ b = 4 – 2a …………… equation 2

Also from equation 1, it is clear that f(x) is also changing its expression at x = 4

Given, f (x) is continuous everywhere

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 230

∴ 8a + 5b = 14 ……………….Equation 3

Putting value of a from equation 2 to equation 3

∴ 8a + 5(4–2a) = 14

⇒ 2a = 6

∴ a = 6/2

= 3

∴ b = 4 – 2×3 = –2

Thus, a = 3 and b = –2

8. If RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 231 for x ≠ π/4, find the value which can be assigned to f (x) at x = π/4 so that the function f (x) becomes continuous everywhere in [0, π/2].

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 232 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 233

Function is defined for [0, π] and we need to find the value of f(x) so that it is continuous everywhere in its domain (domain = set of numbers for which f is defined)

As we have expression for x ≠ π/4, which is continuous everywhere in [0, π], so

If we make it continuous at x = π/4 it is continuous everywhere in its domain.

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 234
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 235
RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 236

Solution:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 237 

Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 238

Function is changing its nature (or expression) at x = 2, so we need to check its continuity at x = 2 first.

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 239

Clearly, LHL = RHL = f (2)

∴ Function is continuous at x = 2

Let c be any real number such that c > 2

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 240

∴ f (x) is continuous everywhere for x > 2.

Let m be any real number such that m < 2

∴ f (m) = 2m – 1 [using equation 1]

RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Image 241

∴ f (x) is continuous everywhere for x < 2.

Hence, we can conclude by stating that f(x) is continuous for all Real numbers

Also, access RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity

Exercise 9.1 Solutions

Exercise 9.2 Solutions

Leave a Comment

Your email address will not be published. Required fields are marked *