NCERT Solutions for Class 11 Maths Chapter 12 Exercise 12.1 Limits and Derivatives - PDF Download

NCERT solutions for class 11 maths chapter 12 exercise 12.1 Limits and Derivatives are based on Limits and their algebra. The first part introduces limits and their intuitive understanding. Ex 12.1 class 11 maths chapter 12 NCERT solutions teaches students how to use limits in polynomials, rational functions and trigonometrics. 

Ex 12.1 class 11 maths chapter 12 has a total of 32 questions, Some of the questions are simply formula based, while others questions necessitate students to utilize their mathematical abilities due to their level of difficulty. Students can solve all the questions easily if they study the theoretical concepts in a structured manner. 

Ex 12.1 class 11 maths ch 12 NCERT solutions are provided in PDF format in which concepts of limits have been explained in detail by our subject experts of eSaral. Downloading these PDFs will provide you with a proper understanding of concepts to perform effectively in final examinations. Download the free PDF from the link given below.

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Topics Covered in Exercise 12.1 Class 11 Mathematics Questions

Ex 12.1 class 11 maths chapter 12 Limits and Derivatives is based on topics like intuitive idea of derivatives, limits, algebra of limits, limits of polynomials and rational functions and limits of trigonometric functions. 

1. Intuitive Idea of Derivatives

A derivative is a measure of the rate of change of a particular function or quantity in comparison to other values. The equation for a derivative may take the form of

$\lim _{a \rightarrow 0} \frac{f(x+a)-f(x)}{a}$

2. Limits - A limit is defined as the value that a function (f(x)) reaches as the limit reaches some value. Limits define the integration, the integral calculus and the continuity of a function.

The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit.

Limit of a function at a point is the common value of the left and right hand limits, if they coincide.

If f(y) is a function, then the limit of the function can be shown as;

                 lim_y→c

This is a general expression of a limit, where the value c can be any constant.

• Algebra of limits

Theorem 1: Let f and g be two functions such that both lim_{x→ a} f(x) and lim_{x→ a}g(x) exist. Then-

 (i) Limit of sum of two functions is sum of the limits of the functions.

                  lim_{x→ a} [f(x) + g (x)] = lim_{x→ a}  f(x) + lim_{x→ a} g(x).

(ii) Limit of difference of two functions is the difference of the limits of the functions.

                  lim_{x→ a} [f(x) – g(x)] = lim_{x→ a} f(x) – lim_{x→ a} g(x).

(iii) Limit of product of two functions is product of the limits of the functions.

                  lim_{x→ a} [f(x) . g(x)] = lim_{x→ a} f(x). lim_{x→ a} g(x).

(iv) Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero).

                    $\lim _{\mathrm{X} \rightarrow \mathrm{a}} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}$

(V) $\lim _{\mathrm{x} \rightarrow \mathrm{a}}[(\lambda . f)(x)]=\lambda \cdot \lim _{x \rightarrow a} f(x)$

• Limits of polynomials and rational functions

Theorem 2: For any positive integer n, 

         $\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n a^{n-1}$

         $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$

         $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=0$

3. Limits of Trigonometric Functions

The following theorems about general functions are useful when calculating the limits of certain trigonometric functions.

Theorem 3: Let f and g be two real valued functions with the same domain such that f (x) ≤ g( x) for all x in the domain of definition, For some a, if both lim_x→a f(x) and lim_x→a g(x) exist, then lim_x→a f(x) ≤ lim_x→a g(x).

Theorem 4: (Sandwich Theorem): Let f, g and h be real functions such that f(x) ≤ g( x) ≤ h(x) for all x in the common domain of definition. For some real number a, if lim_x→a f(x) = l = lim_x→a h(x), then lim_x→a g(x) = l

Theorem 5: The following are two important limits. 

(I) $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$       (II)  $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=0$

Tips for Solving Exercise 12.1 Class 11 Chapter 12 Limits and Derivatives

Here, our subject experts of eSaral have combined some useful tips to solve questions of ex 12.1 class 11 maths chapter 12 Limits and Derivatives.

1. NCERT solutions class 11 maths chapter 12 exercise 12.1 is a great exercise to understand limits and how functions work under different conditions defined by limits so you should comprehend with the concepts to solve questions of ex 12.1.
2. Ex 12.1 class 11 maths chapter 12 describes some important formulas related to limits which is essential to learn before solving ex 12.1.
3. Ex 12.1 has questions which are based on theorems. Students must study these theorems to solve questions.

Importance of Solving Ex 12.1 Class 11 Maths Chapter 12 Limits and Derivatives

There are numerous benefits of solving questions of ex 12.1 class 11 maths chapter 12 Limits and Derivatives. Here, we have provided some of them which you can check below.

1. Ex 12.1 class 11 maths ch 12 offers an in-depth and comprehensive introduction to the fundamental concepts of limits, which helps in understanding the questions of exercise 12.1.
2. Class 11 maths chapter 12 ex 12.1 Limits and theorems are presented in a simple and comprehensible format, making it suitable for students of all mathematical abilities.
3. The chapter offers sequential explanations of the concepts presented, as well as examples and exercise questions to assist students in understanding the concepts presented in the chapter.
4. NCERT solutions PDF for chapter 12 ex 12.1 will assist students to cross check their answers.
   

Frequently Asked Questions 

Question 1. What are the key topics in NCERT solutions class 11 maths chapter 12 ex 12.1?

Answer 1. NCERT solutions class 12 maths chapter 12 ex 12.1 has essential topics which you can find below:

1. Intuitive Idea of Derivatives 
2. Limits
3. Algebra of limits
4. Limits of polynomials and rational functions
5. Limits of Trigonometric Functions

Question 2. How are limits important in NCERT solutions class 11 maths chapter 12 ex 12.1?

Answer 2. These limits help us figure out the nearest values of the function that is part of the calculation.