RD Sharma Solutions For Class 12 Maths Exercise 9.2 Chapter 9 Continuity

This exercise mainly focuses on continuity on an interval of a given function. Highly experienced BYJU’S subject experts having a vast knowledge of concepts prepare the answers, which match the understanding ability of the students. RD Sharma Solutions for Class 12 Maths Chapter 9 Continuity Exercise 9.2 are provided here. Let us have a look at some of the topics that are discussed in this exercise.

  • 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 Chapter 9 – Continuity Exercise 9.2:-Download PDF Here

RD Sharma Class 12 Maths Solutions Chapter 9 Continuity Exercise 9.2
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Access another exercise of RD Sharma Solutions For Class 12 Chapter 9 – Continuity

Exercise 9.1 Solutions

Access answers to Maths RD Sharma Solutions For Class 12 Chapter 9 – Continuity Exercise 9.2

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

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