Limits And Derivatives Class 11th Notes - Free NCERT Class 11 Maths Chapter 13 notes - Download PDF

Class 11 Math chapter 13 notes are regarding Limits and their derivatives. In our daily lives, we see the motion of vehicles, speedometers, and changing temperatures, which involve changing quantities. These changes are studied using limits and derivatives. So, what is a limit? A limit helps us understand the behavior of a function as it approaches a particular value. A derivative tells us the rate of change of a function. For example, speed is the derivative of distance with respect to time. Derivatives are based on limits. They form the foundation of calculus and are also used in physics. NCERT Class 11 Maths Chapter 13 contains the following topics: Limits, representation, left-hand limits, right-hand limits, algebra of limits, algebra of polynomials and rational functions, trigonometric functions theorem, standard limit formulas of limits, Derivatives, representation, algebra of derivatives, algebra of polynomials, standard derivative formulas.

This article on NCERT notes Class 11 Maths Chapter 13 Continuity and Differentiability offers well-structured NCERT notes to help the students to prepare for their exams in a scheduled manner. These notes of Class 11 Maths Chapter 13 Continuity and Differentiability are made by the Subject Matter Experts strictly based on the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. NCERT solutions for class 11 maths and NCERT solutions for other subjects and classes can be downloaded from NCERT Solutions.

NCERT CLASS 11 CHAPTER 13 NOTES

Limits

Let $f$ be a function defined in a domain which we take to be an interval, say, I. We shall study the concept of the limit of $f$ at a point ' $a$ ' in I.

We say $\underset{x\to a^{-}}{lim}f(x)$ is the expected value of $f$ at $x=a$ given the values of $f$ near to the left of $a$. This value is called the left-hand limit of $f$ at $a$.

We say $\underset{x\to a^{+}}{lim}f(x)$ is the expected value of $f$ at $x=a$ given the values of $f$ near to the right of $a$. This value is called the right-hand limit of $f$ at $a$.
If the right and left-hand limits coincide, we call the common value the limit of $f$ at $x=a$ and denote it by $\underset{x\to a}{lim}f(x)$.

If y=f(x) is a function where x=a, then it cannot take exact values, but closure to a, then limits are used to obtain a unique number when x tends to a or some value that is definite.

Mathematical representation: $\underset{x\to a}{lim}f(x)$

Left-Hand Limit:

If a function has values close to x=a on the left hand, then the unique value is given by:

$f\left(a^{-}\right)=\underset{x\to a^{-}}{lim}f(x)=\underset{h\to 0}{lim}f(a-h)$

Right-Hand Limit:

If a function that is given has values close to x=a on the right hand, then the unique value is given by:

$f\left(a^{+}\right)=\underset{x\to a^{+}}{lim}f(x)=\underset{h\to 0}{lim}f(a+h)$

Algebra of Limits

Theorem 1:

If f and g are two functions where both limits lim_x→a f(x) and lim_x→a g(x) exist.

i) The limit of the sum of two functions is equal to the limit of the individual sums of each function.

lim_x→a [f(x)+g(x)] = lim_x→a f(x) + lim_x→a g(x)

ii) The limit of difference between two functions is equal to the limit of individual differences of each function.

lim_x→a [f(x)-g(x)] = lim_x→a f(x) - lim_x→a g(x)

iii) The limit of the product of two functions is equal to the limit of the individual product of each function.

lim_x→a [f(x).g(x)] = lim_x→a f(x) . lim_x→a g(x)

iv) The limit of the quotient of two functions is equal to the quotient of the limit of each function.

lim_x→a f(x)g(x) = lim_x→a f(x)xag(x)

v)lim_x→a (λ.f(x)) = λ . lim_x→a f(x)

Limits of polynomials and rational functions
If $f$ is a polynomial function, then $\underset{x\to a}{lim}f(x)$ exists and is given by

$\underset{x\to a}{lim}f(x)=f(a)$

An important limit, which is very useful and used in the sequel, is given below:

$\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

Remark: The above expression remains valid for any rational number provided that ' $a$ ' is positive.
Limits of trigonometric functions
To evaluate the limits of trigonometric functions, we shall make use of the following limits, which are given below:

(i) $\underset{x\to 0}{lim}\frac{\sin x}{x}=1$
(ii) $\underset{x\to 0}{lim}\cos x=1$
(iii) $\underset{x\to 0}{lim}\sin x=0$

Some Standard Limit Functions to Remember:

1. lim_x→a [(xⁿ-aⁿ)/(x-a)]= n.a^n-1

2. lim_x→0 [(sin x)/x] = 1

3. lim_x→0 [(tan x)/x] = 1

4. lim_x→0 [e^x/x] = 1

5. lim_x→0 [a^x-1)/x] = log_ea

6. lim_x→0 [(log(1+x))/x] = 1

Limits of Polynomial and Rational Functions:

1. lim_x→a x = a

2. lim_x→a xⁿ=aⁿ

3. When f(x) = g(x)/h(x) then

lim_x→a f(x) = lim_x→a [g(x)/h(x)] = [lim_x→a g(x)] / [lim_x→a h(x)] =g(a)/h(a)

Limits of Trigonometric Functions:

Theorem 2:

If f(x) and g(x) are two real-valued functions with the same domain satisfying

f(x)≤g(x) and limit of two functions lim_x→a f(x) and lim_x→a g(x) exists then

lim_x→a f(x) ≤ lim_x→a g(x)

Theorem 3: (Sandwich Theorem)

If f(x), g(x), hand (x) are real-valued functions with the same domain satisfying

f(x)≤g(x)≤h(x), if lim_x→a f(x) = l = lim_x→a h(x) then lim_x→a g(x) =l

Derivatives

1. If f is a real-valued function and a is the domain, then the derivative of f

lim_h→0 (f(a+h)-f(a))/h exists.

The derivative of f(x) is denoted by f’(a).

Therefore : f’(a) = lim_h→0 (f(a+h)-f(a))/h

2. If f is a real-valued function, then limits exist for

f(x) = lim_h→0 (f(x+h)-f(x))/h

Algebra of Derivatives of Functions

1. The derivative of the sum of two functions is equal to the sum of the individual derivatives of each function.

d/dx [f(x)+g(x)] = (d/dx) f(x) + (d/dx) g(x)

2. The derivative of the difference of two functions is equal to the derivative of the individual difference of each function.

(d/dx) [f(x)-g(x)] = (d/dx) f(x) - (d/dx) g(x)

3. The derivative of the product of two functions is equal to the derivative of the individual product of each function.

(d/dx) [f(x).g(x)] = (d/dx) f(x) . g(x)+f(x). (d/dx) g(x)

4. The derivative of the quotient of two functions is equal to the quotient of the derivative of each function.

(d/dx) [f(x)/g(x)] = [(d/dx) f(x) . g(x)-f(x). (d/dx) g(x)] / (g(x))²

Derivative of Polynomial and Trigonometric Functions:

If f(x) is said to be a polynomial function such that

f(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ……………….+ a₁x+a₀

Then the derivative of the function is given by:

(d/dx) (f(x)= na_nx^n-1+ (n-1)a_n-1x^n-2+(n-2)a_n-2x^n-3+..............+2a₂x+a₁

Some standard Derivatives:

1. $(d/dx)\left(x^n\right)=n{x}^{n-1}$
2. $(d/dx)(\sin x)=\cos x$
3. $(d/dx)(\cos x)=-\sin x$
4. $(d/dx)(\tan x)={\sec }^{2}x$
5. $(d/dx)(\cot x)=-{\mathrm{cosec}}^{2}x$
6. $(d/dx)(\sec x)=\sec x\cdot \tan x$
7. $(d/dx)(\sec x)=-\mathrm{cosec}x\cdot \cot x$
8. $(d/dx)\left(a^x\right)=a^x{\log }_{e^a}$
9. $(d/dx)\left(e^x\right)=e^x$
10. $(d/dx)({\log }_{e}x)=1/x$

With this topic, we conclude the NCERT Class 11 chapter 13 notes.

Importance of NCERT Class 11 Maths Chapter 13 Notes:

NCERT Class 11 Maths Chapter 13 notes will be very helpful for students to score good marks in their 11th class exams and help in building a strong base for competitive exams. Also, it is the basic of differentiation to be studied in class 12th.

The CBSE Class 11 Maths Chapter 13 will help to understand the formulas, statements, and rules with their conditions in detail. You can get compact knowledge as a document in the Limits and Derivatives Class 11 chapter 13 pdf download.

NCERT problems are very important in order to perform well in exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives. Students are advised to go through the NCERT Class 11 Maths Chapter 13 Notes before solving the questions.

These NCERT notes are very useful for boosting your exam preparation and quick revision.

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