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NCERT Solutions for Class 11 Maths Chapter 13 : Limits and Derivatives

Q: Q1. Define Limits and what are its significance?

Answer: In mathematics, a limit is defined as a value that is approached as input by a function and produces some output or result. It is only because of its limitations, we can study a number from a distance. That is, we can look at the points around it to gain a better grasp of the given value.

Q: Q2. How many exercises are there in the Limits and Derivatives Class 11 NCERT Solutions?

Answer: There are 73 sums in Class 11 Maths Chapter 13 Limits and Derivatives, with 23 being simple formula-based questions, 20 being medium level, and 30 being challenging. These are divided into three exercises, one of which is miscellaneous and contains higher-order problems. The solutions to the exercise-specific problems are also accessible in PDF format, with the goal of assisting students in performing well in the yearly exam.

Q: Q3. What are the key subtopics in Limits and Derivatives Class 11 Maths Chapter 13 that could be tested?

Answer: NCERT Solutions Class 11 Maths Chapter 13 begins with a brief overview of the calculus concepts. The NCERT Solutions Class 11 Maths Chapter 13 focuses on calculating the limits of a function as well as the algebra of limits and derivatives. Students can now study and stay up to date on the latest CBSE syllabus by using the NCERT Solutions, which are available in PDF format.

Q: Q4. Is it necessary to practice all of the questions in Limits and Derivatives Class 11 NCERT Solutions?

Answer: Limits and Derivatives Class 11 NCERT Solutions are well-designed to help students study core calculus concepts easily. All 3 exercises in the NCERT Solutions Class 11 Maths Limits and Derivatives assignments cover various topics of the course. It thoroughly covers all essential principles, allowing students to grasp the concepts. All of the exercises, examples, and practice problems have been meticulously designed to help students gain a rapid and effective comprehension of the various topics.

Limits and Derivatives Class 11 Maths Chapter 13 deals with the study of Calculus which is a field of mathematics that studies the change in the value of a function as the points in the domain change. Class 11 Maths Chapter 13 explains the key concepts of how to use limits, derivatives and the algebra behind derivatives. Class 11 Maths Chapter 13 Limits and Derivatives will help students to gain a thorough understanding of this concept and its practical applications. The expert faculty of Toppr produced these solutions to assist students with their first-term exam preparations.

Limits and Derivatives Class 11 Chapter 13 also covers topics such as the right-hand limit, left-hand limit, algebra of limits, limits of trigonometric functions, derivative formula, algebra of derivatives of functions, and polynomial and trigonometric function derivatives. Students will quickly learn about these concepts if they practice Chapter 13 Maths Class 11 NCERT Solutions of Limits and Derivatives. All of these solutions have been created with the new CBSE pattern in mind, ensuring that students have a thorough understanding of their exams. It goes over all major concepts in depth, allowing students to better understand the ideas.

Chapter 13 Maths Class 11 Questions and Answers can help you get good grades in tests and properly prepare for all of the important concepts. Several examples provided in the solutions will assist students with a better understanding of Limits and Derivatives. Limits and Derivatives Class 11 NCERT Solutions Chapter 13 is available below in pdf format, and a few solutions are also included in the exercises. These solutions explain the topics covered with examples so that students can easily relate to the notion being discussed.

Table of Content

Introduction

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Access NCERT Solutions for Class 11 Maths Chapter 13 : Limits and Derivatives

Exercise 13.1

Question 1

Evaluate the Given limit:

x \to 3 lim (x + 3)

Easy

Solution

Verified by Toppr

x \to 3 lim (x + 3) = 3 + 3 = 6

Note: Given limit is not of indeterminate form

Question 2

Evaluate the Given limit:

x \to π lim (x - \frac{2 2}{7})

Easy

Solution

Verified by Toppr

x \to π lim (x - \frac{2 2}{7}) = (π - \frac{2 2}{7}) \neq = 0

Question 3

Evaluate the Given limit:

r \to 1 lim π r^{2}

Easy

Solution

Verified by Toppr

r \to 1 lim π r^{2} = π (1)^{2} = π

Note: Given limit is not of the indeterminate form.

Question 4

Evaluate:

x \to 3 lim (4 x^{2} + 3)

None of these

Medium

Solution

Verified by Toppr

Given,

x \to 3 lim (4 x^{2} + 3)

Substituting

x = 3

, we get

x \to 3 lim (4 x^{2} + 3) = 4 (3)^{2} + 3 = 4 (9) + 3 = 36 + 3 = 39

Question 5

Evaluate the Given limit:

x \to - 1 lim \frac{x ^{10} + x ^{5} + 1}{x - 1}

Easy

Solution

Verified by Toppr

x \to - 1 lim \frac{x ^{10} + x ^{5} + 1}{x - 1}

= \frac{( - 1 ) ^{10} + ( - 1 ) ^{5} + 1}{- 1 - 1}

= \frac{1 - 1 + 1}{- 2}

= - \frac{1}{2}

Question 6

Evaluate the given limit:

x \to 0 lim \frac{( x + 1 ) ^{5} - 1}{x}

Medium

Solution

Verified by Toppr

x \to 0 lim \frac{( x + 1 ) ^{5} - 1}{x}

Put

x + 1 = y

So,

y \to 1

x \to 0

= y \to 1 lim \frac{y ^{5} - 1 ^{5}}{y - 1}

= 5 \cdot 1^{5 - 1}

[x \to a lim \frac{x ^{n} - a ^{n}}{x - a} = n a^{n - 1}]

= 5

∴ x \to 0 lim \frac{( x + 5 ) ^{5} - 1}{x} = 5

Question 7

Evaluate the Given limit:

x \to 2 lim \frac{3 x ^{2} - x - 1 0}{x ^{2} - 4}

Medium

Solution

Verified by Toppr

x \to 2 lim \frac{3 x ^{2} - x - 1 0}{x ^{2} - 4} = x \to 2 lim \frac{( x - 2 ) ( 3 x + 5 )}{( x - 2 ) ( x + 2 )}

= x \to 2 lim \frac{3 x + 5}{x + 2} = \frac{3 ( 2 ) + 5}{2 + 2} = \frac{1 1}{4}

Question 8

Evaluate the Given limit:

x \to 3 lim \frac{x ^{4} - 8 1}{2 x ^{2} - 5 x - 3}

Medium

Solution

Verified by Toppr

x \to 3 lim \frac{x ^{4} - 8 1}{2 x ^{2} - 5 x - 3} = x \to 3 lim \frac{( x - 3 ) ( x + 3 ) ( x ^{2} + 9 )}{( x - 3 ) ( 2 x + 1 )}

= x \to 3 lim \frac{( x + 3 ) ( x ^{2} + 9 )}{2 x + 1} = \frac{( 3 + 3 ) ( 3 ^{2} + 9 )}{2 ( 3 ) + 1} = \frac{6 \times 1 8}{7} = \frac{1 0 8}{7}

Question 9

Evaluate the Given limit:

x \to 0 lim \frac{a x + b}{c x + 1}

Easy

Solution

Verified by Toppr

Given,

x \to 0 lim \frac{a x + b}{c x + 1}

= \frac{a ( 0 ) + b}{c ( 0 ) + 1}

= b

Question 10

l i m_{z \to 1} \frac{z ^{1 / 3} - 1}{z ^{1 / 6} - 1}

Hard

Solution

Verified by Toppr

lim_{z \to 1} \frac{z ^{\frac{1}{3}} - 1}{z ^{\frac{1}{6}} - 1}

= lim_{z \to 1} \frac{\frac{z ^{\frac{1}{3}} - 1}{z - 1}}{\frac{z ^{\frac{1}{6}} - 1}{z - 1}}

= \frac{\frac{1}{3} \times 1 ^{\frac{- 2}{3}}}{\frac{1}{6} \times 1 ^{\frac{- 5}{6}}}

since

lim_{x \to a} \frac{x ^{n} - a ^{n}}{x - a} = n a^{n - 1}

= \frac{6}{3} = 2

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NCERT Solutions for Class 11 Maths Chapter 13 : Limits and Derivatives

NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives – Brief Overview

Limits

A limit in mathematics is the value that a function approaches when its input approaches some value. Limits are used to define continuity, derivatives, and integrals in calculus and mathematical analysis. It is only because of its limitations, we can study a number from a distance.

Few Properties of Limits
- lim_x→a c = c, where c is a constant quantity.
- The value of lim_x→ax = a
- Value of lim_x→a bx + c = ba + c
- lim_x→a xⁿ = aⁿ, if n is a positive integer.
- lim_x→0⁻1/x^r = −∞, if r is odd, and
- lim_x→0⁻1/x^r = +∞, if r is even.

Derivatives

In mathematics, a derivative is the rate of change of a function with respect to a variable. Derivatives are essential for solving calculus and differential equation issues. Suppose f is a real-valued function and a is a point in its domain of definition. The derivative of f at a is defined by,

lim_h→0 [ f(a + h) – f (a) / h ]

provided this limit exists. Derivative of f(x) at a is denoted by f′(a).

Related Chapters

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

Frequently Asked Questions on NCERT Class 11 Maths Chapter 13 : Limits and Derivatives

Q1. Define Limits and what are its significance?

Answer: In mathematics, a limit is defined as a value that is approached as input by a function and produces some output or result. It is only because of its limitations, we can study a number from a distance. That is, we can look at the points around it to gain a better grasp of the given value.

Q2. How many exercises are there in the Limits and Derivatives Class 11 NCERT Solutions?

Answer: There are 73 sums in Class 11 Maths Chapter 13 Limits and Derivatives, with 23 being simple formula-based questions, 20 being medium level, and 30 being challenging. These are divided into three exercises, one of which is miscellaneous and contains higher-order problems. The solutions to the exercise-specific problems are also accessible in PDF format, with the goal of assisting students in performing well in the yearly exam.

Q3. What are the key subtopics in Limits and Derivatives Class 11 Maths Chapter 13 that could be tested?

Answer: NCERT Solutions Class 11 Maths Chapter 13 begins with a brief overview of the calculus concepts. The NCERT Solutions Class 11 Maths Chapter 13 focuses on calculating the limits of a function as well as the algebra of limits and derivatives. Students can now study and stay up to date on the latest CBSE syllabus by using the NCERT Solutions, which are available in PDF format.

Q4. Is it necessary to practice all of the questions in Limits and Derivatives Class 11 NCERT Solutions?

Answer: Limits and Derivatives Class 11 NCERT Solutions are well-designed to help students study core calculus concepts easily. All 3 exercises in the NCERT Solutions Class 11 Maths Limits and Derivatives assignments cover various topics of the course. It thoroughly covers all essential principles, allowing students to grasp the concepts. All of the exercises, examples, and practice problems have been meticulously designed to help students gain a rapid and effective comprehension of the various topics.

Related Chapters

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