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Students of class 12 carry a big burden in the form of public exams. Best publications like the RD Sharma books, cater the needs of the students to make them understand every concept clearly. RD Sharma Solution Mathematics is a complicated subject for the class 12 students. The Class 12 RD Sharma Chapter Exercise 8.1 solution plays a key role in making the students understand those concepts in-depth.

**Also Read - **RD Sharma Solution for Class 9 to 12 Maths

- Chapter 8 - Continuity -Ex-8.2
- Chapter 8 - Continuity -Ex-FBQ
- Chapter 8 - Continuity -Ex-MCQ
- Chapter 8 - Continuity -Ex-VSA

Discontinuous

The discontinuity occurs when the LHL and RHL are not equal at a point

Given

We observe,

[LHL at ]

[RHL at ]

Hence, is discontinuous at the

Continuity Excercise 8.1 Question 2

**Answer:****Hint:**

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.**Solution:**

Given,

We observe,

[LHL at ]

[RHL at ]

Also,

Hence, is continuous at .

Continuity Excercise 8.1 Question 3

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe,

[LHL at ]

[RHL at ]

Given

Hence, is continuous at .

Continuity Excercise 8.1 Question 4

Continuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe,

[LHL at ]

[RHL at ]

Given

Hence, is continuous at .

Continuity Excercise 8.1 Question 5

(Discontinuous)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe,

[LHL at ]

[RHL at ]

Given

It is known that for a function is to be continuous at ,

But here

Hence, is discontinuous at .

Continuity Exercise 8.1 Question 6

(Discontinuous)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe,

[LHL at ]

[RHL at ]

Given

It is known that for a function is to be continuous at ,

But here

Hence, is discontinuous at .

Continuity Exercise 8.1 Question 7

(Discontinuous)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

Consider,

Given

Thus, is discontinuous at .

Continuity Exercise 8.1 Question 8

(Discontinuous)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe,

[LHL at ]

[RHL at ]

And

Thus, is discontinuous at .

Continuity Exercise 8.1 Question 9

(Discontinuous)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe

[LHL at ]

[RHL at ]

Thus, is discontinuous at .

Continuity Exercise 8.1 Question 10 (i)

Continuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe,

The limit of function at tends to is equal to the value of function at that point hence it is continuous.

Hence, is continuous at .

Continuity Exercise point 1 Question 10 (ii)

Continuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe

Hence, is continuous at .

Continuity Exercise 8.1 (iii) Question 10

Continuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

Putting , we get

Hence, is continuous at .

Continuity Exercise 8.1 Question 10 (iv)

Discontinuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe,

[Multiplying and dividing the denominator by ]

And

Thus, is discontinuous at .

Discontinuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

Given,

Here,

Thus, is discontinuous at .

Continuity Excercise 8.1 Question 10 (vi)

Continuous at

For a function to be continuous at a point, its LHL and RHL value at that point should be equal.

Given,

We observe

[LHL at ]

[RHL at ]

RHL=LHL

Therefore, is continuos at

Continuity Excercise 8.1 Question 10 (vii)

Discontinuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe

[LHL at ]

[RHL at ]

Thus, is discontinuous at .

Continuity Excercise 8.1 Question 10 (viii)

Continuous

Continuous function must be defined at a point, limit must exist at the point, value of the function at that point must equal the value of the right and left limit at that point.

Given,

We observe

[LHL at ]

[RHL at ]

Thus, is continuous at .

Continuity Exercise 8.1 Question 11

Discontinuous

Discontinuous occurs when the both number is equal to zero of the numerator and denominator or the function is undefined at its limit.

Given,

We observe

[LHL at ]

[RHL at ]

Thus, is discontinuous at .

Continuity Exercise 8.1 Question 12

Continuous

Continuous function must be defined at a point, limit must exist at the point and the value of the function at that point must equal the value of the right hand and left hand limit at that point.

Given,

We observe

[LHL at ]

And

Thus, is continuous at .

Continuity exercise 8.1 question 13

Continuous function must be defined at a point, limit must exist at the point and the value of the function at that point must equal to the value of the right hand or left hand limit at that point.

Given,

We observe

[LHL at ]

[RHL at ]

[Multiplying and dividing the denominator by ]

If is continuous at , then

Continuity Exercise 8.1 Question 14

is discontinuous at the point

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

at

We observe

[LHL at ]

[RHL at ]

Thus, is discontinuous at .

Continuity Exercise 8.1 Question 15

Discontinuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe

[LHL at ]

[RHL at ]

And

Thus, is discontinuous at .

Continuity exercise 8.1 question 16

Continuous

Continuous function must be defined at a point, limit must exist at the point and the value of the function at that point must equal the value of the right hand or left hand limit at that point.

Given,

We observe

[LHL at ]

[RHL at ]

Also

Thus, is continuous at .

Continuity Exercise 8.1 Question 17

Discontinuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

We observe

[LHL at ]

[RHL at ]

Thus, is discontinuous at .

Continuity exercise 8.1 question 18

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

If is continuous at , then

Continuity exercise 8.1 question 19

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Given,

If is continuous at , then

Continuity exercise 8.1 question 20

must be defined. The limit of the approaches the value must exist.

is continuous at

If is continuous at , then

[Multiplying and dividing by ]

Continuity Exercise 8.1 Question 21

must be defined. The limit of the approaches the value must exist.

is continuous at

If is continuous at , then

Continuity Exercise 8.1 Question 22

must be defined. The limit of the approaches the value must exist.

is continuous at

If is continuous at , then

[Multiplying and dividing by ]

Continuity Excercise 8.1 Question 23

must be defined. The limit of the approaches the value must exist.

We observe

[LHL at ]

[RHL at ]

is continuous at , we have

Continuity Excercise 8.1 Question 24

Since, and are not equal, is discontinuous.

Thus, is discontinuous at

must be defined. The limit of the approaches the value must exist.

We observe

(LHL at )

(RHL at )

Since, and are not equal, is discontinuous.

Thus, is discontinuous at , regardless of choice of

Continuity exercise 8.1 question 25

must be defined. The limit of the approaches the value must exist.

If is continuous at , then

......(i)

Putting

From (i)

, is continuous at .

Continuity Exercise 8.1 Question 26

must be defined. The limit of the approaches the value must exist.

Since is continuous at , we have

(LHL at )

RHL at

Since is continuous at , then

is any real number except

Continuity Exercise 8.1 Question 27

must be defined. The limit of the approaches the value must exist.

is continuous at , then

Consider,

[Multiplying and dividing by ]

… (i)

From (i)

must be defined. The limit of the approaches the value must exist.

Solution:

We observe

(LHL at )

(RHL at )

If is continuous at

Continuity Exercise 8.1 Question 29

must be defined. The limit of the approaches the value must exist.

If is continuous at ,

[Multiplying and dividing by ]

Continuity Exercise 8.1 Question 30

must be defined. The limit of the approaches the value must exist.

If is continuous at , then

[Multiplying the denominator of first term by ]

[Multiplying the denominator of second term by ]

Continuity Exercise 8.1 point 12 Question 31

must be defined. The limit of the approaches the value must exist.

If is continuous at ,

[Taking as common]

**Answer:****Hint:**

must be defined. The limit of the approaches the value must exist.**Given:****Solution:**

If is continuous at , then

[Multiplying and dividing by ]

Continuity Exercise 8.1 Question 33

must be defined. The limit of the approaches the value must exist.

If is continuous at , then

[Multiplying and dividing by ]

must be defined. The limit of the approaches the value must exist.

must be defined. The limit of the approaches the value must exist.

If is continuous at , then

Continuity exercise 8.1 question 36 (i)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

If is continuous at ,

Continuity exercise 8.1 question 36 (ii)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

If is continuous at , then

Putting , we get

Continuity Exercise 8.1 Question 36 (iii)

No value of exists.

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

We have

(LHL at )

(RHL at )

Thus no value of exists for which is continuous at

Continuity Exercise 8.1 Question 36 (iv)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Solution:

We have

(LHL at )

(RHL at )

If is continuous at , then

Continuity Excercise 8.1 Question 36 (v)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

We have

(LHL at )

(RHL at )

If is continuous at , then

Continuity Excercise 8.1 Question 36 (vi)

must be defined. The limit of the approaches the value must exist.

If is continuous at , then

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

If is continuous at , then

Continuity Exercise 8.1 Question 36 (viii)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

We have

(LHL at )

(RHL at )

If is continuous at , then

Continuity Exercise 8.1 Question 36 (ix)

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

If is continuous at , then

Continuity Exercise 8.1 Question 37

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

(LHL at )

(RHL at )

If is continuous at and , then

… (i)

… (ii)

On solving equation (i) and (ii), we get

Continuity Exercise 8.1 Question 39 (i)

Continuous

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

We have

(LHL at )

(RHL at )

Also

Now,

(LHL at )

(RHL at )

Also

Hence is continuous at and .

Continuity Excercise 8.1 Question 39

is continuous at

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

Also

Now,

(LHL at )

(RHL at )

Also

Hence is continuous at and .

Continuity Exercise 8.1 Question 40

Discontinuous

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

Thus is discontinuous at .

Continuity Exercise 8.1 Question 41

can be any real number.

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

If is continuous at .

can be any real number.

Continuity Exercise 8.1 Question 42

For any value is Continuous at

must be defined. The limit of the approaches the value must exist.

If is continuous at , then

Putting

When,

At , RHL =

Putting

When,

LHL RHL. Thus, is discontinuous at for any value .

At , LHL=

Putting

When,

At , RHL =

Putting as

When,

LHL RHL

Therefore, for any value of for which is continuous at

Continuity Excercise 8.1 Question 43

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

Also

If is continuous at .

Since is continuous

Thus, is continuous at .

Continuity exercise 8.1 question 44

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

Also

If is continuous at , then

Continuity exercise 8.1 question 45

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

Also

If is continuous at .

Hence, the required value of is .

Continuity exercise 8.1 question 46

must be defined. The limit of the approaches the value must exist.

We have

(LHL at )

(RHL at )

Also

If is continuous at .

Hence the required relationship between

There are a couple of exercises, ex 8.1 and ex 8.2 in this chapter. The concepts that the first exercise, ex 8.1 covers are continuous function, absolute continuous function, continuous probability distribution, absolute continuity of a measure concerning another measure and many more. This first exercise consists of 62 questions to be answered by the students. Looking at the number of questions they need not worry if they have the RD Sharma Class 12th Exercise 8.1 material with them.

Most of the students who utilised the RD Sharma Class 8 solution of Continuity Ex 8.1 book to the best have attained excellent scores in their exams. Even the teachers do not miss to refer to the RD Sharma Class 12th Exercise 8.1 solution book before taking the lessons.

Following are a few advantages that the students attain when they use the RD Sharma Class 12 Solutions Chapter 8 exercise 8.1 solution book:

All the sums in the RD Sharma class 12th Exercise 8.1 reference book are solved in various methods that can be easily grasped by the students.

The fact is that, many teachers pick the homework sums from the RD Sharma Class 12th Exercise 8.1 book. It becomes even easier to recheck the answers or clarify the doubts regarding the homework sums with the help of this book.

The RD Sharma Class 12 Chapter 8 exercise 8.1 and the other RD Sharma books are available at the Career 360 website. Students can also download a copy if they want to use it offline.

Studying with outdated books is useless. The RD Sharma Class 12th Exercise 8.1 are updated according to the latest NCERT textbooks. As the PDF format of these books are available for free of cost at the Career 360 website, the students can make the most of it.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. Who can refer to the RD Sharma Class 12 Chapter 8 ex 8.1 solutions material?

Most of the class 12 students, teachers and tutors who overcome the concept of Continuity use the RD Sharma Class 12 Chapter 8 ex 8.1 solutions.

2. Do the RD Sharma books cost much?

The RD Sharma Class 12 Chapter 8 ex 8.1 book along with the other RD Sharma books are present in the Career 360 website. Anyone can use this resource by downloading it for free.

3. Does the RD Sharma Class 12 Chapter 8 ex 8.1 book contain solution for all the questions asked on the textbook?

Apart from providing worked out solutions for the questions asked in the book, the RD Sharma Class 12 Chapter 8 ex 8.1 contains additional practice questions and answers too.

4. Where can the students find the updated reference materials for class 12 chapter 8?

The updated solution book, RD Sharma Class 12 Chapter 8 ex 8.1 is available at the Career 360 website.

5. What is the most used solution book to crack the public examinations with high scores?

The RD Sharma Class 12 Chapter 8 ex 8.1 is the most used reference material by the previous batch students to crack the public examinations with high scores.

Mar 22, 2023

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