NCERT Exemplar Class 11 Maths Solutions Chapter 13 Limits and Derivatives

# NCERT Exemplar Class 11 Maths Solutions Chapter 13 Limits and Derivatives

Edited By Ravindra Pindel | Updated on Sep 12, 2022 01:56 PM IST

NCERT Exemplar Class 11 Maths Solutions Chapter 13 Limits and Derivatives is drafted by experts teachers of mathematics. The solutions are prescribed by NCERT and using the easiest or rather the most comprehensible methods. To provide reliable and authentic NCERT exemplar Class 11 Maths solutions chapter 13, we aligned to the guidelines of CBSE for the students. The lesson provides for practical application and usage of Limits and Derivatives in various functions of trigonometry, polynomials, and rational numbers.
Also, check - NCERT Class 11 Maths Solutions.

JEE Main Scholarship Test Kit (Class 11): Narayana | Physics WallahAakash Unacademy

Scholarship Test: Vidyamandir Intellect Quest (VIQ)

## NCERT Class 11 Exemplar Maths Solutions Chapter 13 - Question-Wise Solution

Question:1

Evaluate

Question:2

Evaluate

Given that

Question:3

Evaluate $\mathop{\lim }_{h \rightarrow 0}\frac{\sqrt{x+h}-\sqrt{x}}{h}$

$\mathop{\lim }_{h \rightarrow 0}\frac{\sqrt {x+h}-\sqrt {x}}{h}$

$\\=\lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h[\sqrt{x+h}+\sqrt{x}]} \times( \sqrt{x+h}+\sqrt{x}) \ [Rationalizing the denominator] \\\\=\lim _{h \rightarrow 0} \frac{x+h-x}{h[\sqrt{x+h}+\sqrt{x}]} \\\\=\lim _{h \rightarrow 0} \frac{1}{ \sqrt{x+h}+\sqrt{x}}\\\\$
Put limit
$\\= \frac{1}{ \sqrt{x}+\sqrt{x}}\\\\ =\frac{1}{2\sqrt{x}}$

Question:4

Evaluate:

Now put x=x-2 limits change from 0 to 2
$\left[u \sin g \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n \cdot a^{n-1}\right]$

Question:5

Evaluate :

Question:6

Evaluate:

$\left[u \sin g \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n \cdot a^{n-1}\right]$

Question:7

Evaluate:

Question:8

Evaluate:

Question:9

Evaluate:
$\mathop{\lim }_{x \rightarrow \sqrt {2}}\frac{x^{2}-4}{x^{2}+3\sqrt {2}x-8} \\$

\begin{aligned} &=\lim _{x \rightarrow \sqrt{2}} \frac{\left(x^{2}-2\right)\left(x^{2}+2\right)}{x^{2}+4 \sqrt{2} x-\sqrt{2} x-8}\\ &=\lim _{x \rightarrow \sqrt{2}} \frac{(x+\sqrt{2})(x-\sqrt{2})\left(x^{2}+2\right)}{x(x+4 \sqrt{2})-\sqrt{2}(x+4 \sqrt{2})}\\ &=\lim _{x \rightarrow \sqrt{2}} \frac{(x+\sqrt{2})(x-\sqrt{2})\left(x^{2}+2\right)}{(x+4 \sqrt{2})(x-\sqrt{2})}\\&=\lim _{x \rightarrow \sqrt{2}} \frac{(x+\sqrt{2})\left(x^{2}+2\right)}{x+4 \sqrt{2}}\\ &\text { Put limit }\\ &=\frac{(\sqrt{2}+\sqrt{2})(2+2)}{\sqrt{2}+4 \sqrt{2}}\\&=\frac{2 \sqrt{2} \times 4}{5 \sqrt{2}}=\frac{8}{5} \end{aligned}

Question:10

Evaluate:

Let us apply LH rule i.e. L.Hospita'srule to this question

Question:11

Evaluate:

Question:12

Evaluate:
$\mathop{\lim }_{x \rightarrow -3} \left( \frac{x^{3}+27}{x^{5}+243} \right)$

\begin{aligned} &=\lim _{x \rightarrow -3} \frac{\frac{x^{3}+(3)^{3}}{x+3}}{\frac{x^{3}+(3)^{3}}{x+3}} \text { [Dividing the numerator and denominator by } \left.x+3\right]\\ &=\frac{\lim _{x \rightarrow -3}\left(\frac{x^{3}+(3)^{3}}{x+3}\right)}{\lim _{x \rightarrow -3}\left(\frac{x^{3}+(3)^{5}}{x+3}\right)}\left[\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}\right]\\ &=\frac{\lim _{x \rightarrow -3}\left(x^2-3x+9\right)}{\lim _{x \rightarrow -3}\left( x^4 - 3x^3 + 9x^2 - 27x + 81\right)} \\ &=\frac{9+9+9}{81+81+81+81+81}=\frac{3 \times 9}{5 \times 81 }=\frac{1}{5 \times 3}=\frac{1}{15} \end{aligned}

Question:13

Evaluate:

$\\ =\mathop{\lim }_{x \rightarrow \frac{1}{2}} \left( \left( \frac{1}{2x+1} \right) * \left( \frac{12x^{2}+8x-6x-4}{2x-1} \right) \right) \\ \\ =\mathop{\lim }_{x \rightarrow \frac{1}{2}} \left( \left( \frac{1}{2x+1} \right) * \left( \frac{ \left( 3x+2 \right) \left( 4x-2 \right) }{2x-1} \right) ~ \right) \\ \\ =\mathop{\lim }_{x \rightarrow \frac{1}{2}} \left( \left( \frac{1}{2x+1} \right) *2 * \left( 3x+2 \right) ~ \right) =\frac{1}{2 *\frac{1}{2}+1} *2 * \left( 3 *\frac{1}{2}+2 \right) =\frac{7}{2} \\ \\$

Question:14

Evaluate: Find ‘n’, if

Question:15

Evaluate:

Question:16

Evaluate:

Question:17

Evaluate:

Given that

Question:18

Evaluate:

Question:19

Evaluate:

Question:20

Evaluate:

Question:21

Evaluate:

Question:22

Evaluate:

Question:23

Evaluate:

Question:24

Evaluate:

Question:25

Evaluate:

Question:26

Evaluate:

Question:27

Evaluate:

Question:28

Evaluate: If

Question:31

Differentiate each of the functions w.r. to x in

Given that

Applying product rule of differentiation we get

Question:32

Differentiate each of the functions w.r. to x in

y=

Now applying the concept of chain rule

Question:33

Differentiate each of the functions w.r. to x in

Given that

Applying division rule of differentiation that is

Question:34

Given that

Applying division rule of differentiation that is

$\\ \frac{dy}{dx}=\frac{\sin x\frac{d}{dx} \left( x^{5} \right) -x^{5}\frac{d}{dx} \left( sinx \right) }{\sin ^{2}x}-\frac{d}{dx} \left( \cot x \right) \\ \\ =\frac{5x^{4}\sin x-x^{5}\cos x}{\sin ^{2}x}+\mathrm{cosec} ^{2}x \\ \\$

Question:35

Differentiate each of the functions w.r. to x in

Applying division rule of differentiation that is

Given that

Applying division rule of differentiation that is

Question:37

Differentiate each of the functions w.r. to x in

Given that

Applying division rule of differentiation that is

Question:38

Differentiate each of the functions w.r. to x in

Given that

Applying the concept of chain rule

Question:39

Differentiate each of the functions w.r. to x in

This question will involve the concept of both chain rule and product rule

Given that

Applying product rule of differentiation

Question:40

Differentiate each of the functions w.r. to x in

This question will involve the concept of both chain rule and product rule

Question:41

Differentiate each of the functions w.r.to x in

The question involves the concept of chain rule

Question:42

Differentiate each of the functions w.r. to x in

The question involves the concept of chain rule

Question:45

Differentiate each of the functions with respect to ‘x’
Differentiate using first principle

Expanding by binomial theorem and rejecting the higher powers of

Question:49

Question:51

Evaluate each of the following limits
Show that
does not exist.

Hence, the limit doesnot exist

Question:54

Hence, the answer is option C

Question:55

Hence, the answer is option A

Question:56

Hence, the answer is option A

Question:57

Hence, the answer is option B

Question:58

Hence, the answer is option A

Question:59

Hence, the answer is option C

Question:60

Hence, the answer is option C

Question:61

Choose the correct answer out of 4 options given against each Question

is
A. 3
B. 1
C. 0
D.

$\\ \mathop{\lim }_{x \rightarrow \frac{ \pi }{4}} \left( \frac{\sec ^{2}x-2}{\tan x-1} \right)\\ =\mathop{\lim }_{x \rightarrow \frac{ \pi }{4}} \left( \frac{1+\tan ^{2}x-2}{\tan x-1} \right) \\=\mathop{\lim }_{x \rightarrow \frac{ \pi }{4}} \left( \frac{\tan ^{2}x-1}{\tan x-1} \right)\\ =\mathop{\lim }_{x \rightarrow \frac{ \pi }{4}} \left( \tan x+1 \right)\\ =2 \\ \\$
Hence, the answer is option D

Question:62

Choose the correct answer out of 4 options given against each Question
is

A.
B.
C. 1
D. None of these

Hence, the answer is option B

Question:63

Limit doesn’t exist
Hence, the answer is option D

Question:64

Choose the correct answer out of 4 options given against each Question

is

A. 1
B. –1
C. does not exist
D. None of these

Question:65

Hence, the answer is option D

Question:66

Hence, the answer is option B

Question:67

Hence, the answer is option B

Question:68

Hence, the answer is option D

Question:69

Hence, the answer is option A

Question:70

Hence, the answer is option A

Question:71

Choose the correct answer out of 4 options given against each Question

If then is

A. –2
B. 0
C.
D. does not exist

Hence, the answer is option A

Question:72

Choose the correct answer out of 4 options given against each Question

If then is

A. cos 9
B. sin 9
C. 0
D. 1

Hence, the answer is option A

Question:73

Choose the correct answer out of 4 options given against each Question

If then f’(1) is equal to

A. 1/100
B. 100
C. does not exist
D. 0

Hence, the answer is option B

Question:74

Choose the correct answer out of 4 options given against each Question

If for some constant ‘a’, then f’(a) is

A. 1
B. 0
C. does not exist
D. 1/2

Hence, the answer is option B

Question:75

Choose the correct answer out of 4 options given against each Question
If , then f’(1) is equal to

A. 5050
B. 5049
C. 5051
D. 50051

Hence, the answer is option A

Question:76

Hence, the answer is option D

Question:77

Fill in the blanks
If

Question:78

Fill in the blanks
then m = ________

$\\ \mathop{\lim }_{x \rightarrow 0}\sin mx\cot \frac{x}{\sqrt {3}}=2 \\ \\ \mathop{\lim }_{x \rightarrow 0}mx * \left( \frac{\sin mx}{mx} \right) * \left( \frac{\frac{x}{\sqrt {3}}}{\tan \frac{x}{\sqrt {3}}} \right) * \left( \frac{\sqrt{3}}{x} \right) =2 \\ \\ \mathop{\lim }_{x \rightarrow 0}mx * \left( \frac{\sqrt {3}}{x} \right) =2 \\ \\ \sqrt {3}m=2 \\ \\ m=\frac{2}{\sqrt {3}}=\frac{2\sqrt {3}}{3} \\ \\$

## More About NCERT Exemplar Solutions for Class 11 Maths Chapter 13

NCERT Exemplar Class 11 Maths chapter 13 solutions is a great way for students to learn and understand the concept of limits and derivatives through easier methods prescribed by the experts and develop their base for the same.

NCERT Exemplar Class 11 Maths solutions chapter 13 PDF download is useful for students to read offline. Use an online webpage to PDF tool for this. These solutions make learning more convenient and includes the detailed study of methods and guidelines of CBSE and NCERT.

The NCERT Exemplar solutions for Class 11 Maths chapter 13 will provide access to efficient and carefully drafted solutions to students to aid preparation and learning process for a better outcome.

### Topics and Subtopics in Class 11 Maths NCERT Exemplar Solutions Chapter 13

• 13.1 Introduction
• 13.2 Intuitive Idea of Derivatives
• 13.3 Limits
• 13.3.1 Algebra of Limits
• 13.3.2 Limits of polynomials and rational functions
• 13.4 Limits of Trigonometric function
• 13.5 Derivatives
• 13.5.1 Algebra of Derivative of functions
• 13.5.2 Derivatives of polynomials and trigonometric functions
• 13.6 Miscellaneous Examples

### What will the Students Learn in NCERT Exemplar Class 11 Maths Chapter 13 solutions?

• NCERT Exemplar Class 11 Maths solutions chapter 13 covers the really important topic of limits and derivatives of any function which is a very important concept for mathematics as well as physics.

• The students will learn about calculating the limits of any function which is important for calculus and mathematical analysis that are further used to define integrals, derivatives, and continuity of different functions.

## NCERT Exemplar Class 11 Maths Solutions Chapter-Wise

 Chapter 1 Sets Chapter 2 Relations and Functions Chapter 3 Trigonometric Functions Chapter 4 Principle of Mathematical Induction Chapter 5 Complex Numbers and Quadratic Equations Chapter 6 Linear Inequalities Chapter 7 Permutations and Combinations Chapter 8 Binomial Theorem Chapter 9 Sequences and Series Chapter 10 Straight lines Chapter 11 Conic Sections Chapter 12 Introduction to Three Dimensional Geometry Chapter 14 Mathematical Reasoning Chapter 15 Statistics Chapter 16 Probability

## Important Topics To Cover From NCERT Exemplar Class 11 Maths Solutions Chapter 13

Some of the important topics for students to review are as follows:

• The students will learn about the limits of different functions from NCERT exemplar solutions for Class 11 Maths chapter 13

• The students will be able to define the limits and derivatives of different trigonometric, polynomial and rational number functions.

• The NCERT exemplar Class 11 Maths solutions chapter 13 covers various solved examples along with solutions for better understanding and learning of different concepts.

• The students should practice the application of different formulas provided in the chapter along with solutions and solved examples, take help from Class 11 Maths NCERT exemplar solutions chapter 13.

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Check Chapter-Wise NCERT Solutions of Book

 Chapter-1 Sets Chapter-2 Relations and Functions Chapter-3 Trigonometric Functions Chapter-4 Principle of Mathematical Induction Chapter-5 Complex Numbers and Quadratic equations Chapter-6 Linear Inequalities Chapter-7 Permutation and Combinations Chapter-8 Binomial Theorem Chapter-9 Sequences and Series Chapter-10 Straight Lines Chapter-11 Conic Section Chapter-12 Introduction to Three Dimensional Geometry Chapter-13 Limits and Derivatives Chapter-14 Mathematical Reasoning Chapter-15 Statistics Chapter-16 Probability

### NCERT Exemplar Class 11 Solutions

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Read more NCERT Solution subject wise -

Also, read NCERT Notes subject wise -

Also Check NCERT Books and NCERT Syllabus here:

NCERT exemplar Class 11 Maths solutions chapter 13 pdf download can be accessed by using the webpage to PDF tool.

2. Is the chapter important from the perspective of Board and competitive exams?

Yes, the chapter has prominent weightage in both Board and competitive exams and helps to understand the basics terms and methods important from an examination perspective.

3. Which are the important topics that students must cover in this chapter?

The students must cover limits for trigonometric functions and also for polynomials. Derivatives also make a base for future learning in this NCERT exemplar Class 11 Maths chapter 13 solutions.

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

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 Option 1) Option 2) Option 3) Option 4)

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In the reaction,

 Option 1)   at STP  is produced for every mole   consumed Option 2)   is consumed for ever      produced Option 3) is produced regardless of temperature and pressure for every mole Al that reacts Option 4) at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, will contain 0.25 mole of oxygen atoms?

 Option 1) 0.02 Option 2) 3.125 × 10-2 Option 3) 1.25 × 10-2 Option 4) 2.5 × 10-2

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