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Edited By Ramraj Saini | Updated on Mar 22, 2022 05:29 PM IST

Class 11 Math chapter 13 notes are regarding Limits and their derivatives. In chapter 13 we will be going through the Limits and derivations concepts in Limits and Derivatives Class 11 notes. This Class 11 Maths chapter 13 notes 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.

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This Story also Contains

- NCERT CLASS 11 CHAPTER 13 NOTES
- Limits:
- Algebra of Limits:
- Some Standard Limit Functions to Remember:
- Limits of Polynomial and Rational Functions:
- Limits of Trigonometric Functions:
- Derivatives:
- Algebra of Derivative of Functions:
- Derivative of Polynomial and Trigonometric Functions:
- Significance of NCERT Class 11 Maths Chapter 13 Notes:
- NCERT Class 11 Notes Chapter Wise.

NCERT Class 11 Math chapter 13 notes also contain formulas and principles that are to be remembered for the implementation in problems. By going through this document students can cover all the topics that are in the NCERT notes for Class 11 Math chapter 13 textbook. It also contains examples and FAQ’s that are asked questions by students. Every concept that is in CBSE Class 11 Maths chapter 13 notes is explained here in a simple and way that can reach students easily. All these concepts can be downloaded from Class 11 Maths chapter 13 notes pdf download, Limits and Derivatives Class 11 notes, Class 11 Limits and Derivatives notes pdf download.

**Also, students can refer,**

- NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives
- NCERT Exemplar Class 11 Maths Chapter 13 Limits and Derivatives

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

Mathematical representation:

**Left-Hand Limit:**

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

**Right-Hand Limit: **

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

**Theorem 1:**

If f and g be two functions where both limits lim_{x→a} f(x) and lim_{x→a} g(x) exist.

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

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 equal to limit of individual difference of each function.

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

iii) 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) limit of the quotient of two functions is equal to the quotient of limit of each individual 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)

lim

_{x→a}[(x^{n}-a^{n})/(x-a)]= n.a^{n-1}lim

_{x→0}[(sin x)/x] = 1lim

_{x→0}[(tan x)/x] = 1lim

_{x→0}[e^{x}/x] = 1lim

_{x→0}[a^{x}-1)/x] = log_{e}alim

_{x→0}[(log(1+x))/x] = 1

lim

_{x→a}x = alim

_{x→a}x^{n}=a^{n}When f(x) = g(x)/h(x) then

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Download EBooklim_{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)

**Theorem 2:**

If f(x) and g(x) be two real valued functions with 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), h(x) be real valued functions with 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 .

If f is a real-valued function and a is domain then derivative of f

lim_{h→0} (f(a+h)-f(a))/h exists.

Derivative of f(x) is denoted by f’(a).

Therefore : f’(a) = lim_{h→0} (f(a+h)-f(a))/h

If f is real valued function then limits exists for

f(x) = lim_{h→0} (f(x+h)-f(x))/h

Derivative of the sum of two functions is equal to the derivative of individual sums of each function.

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

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)

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)

Derivative of the quotient of two functions is equal to the quotient of the derivative of each individual function.

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

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

f(x) = a_{n}x^{n }+ a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ……………….+ a_{1}x+a_{0}

Then derivative of the function is given by:

(d/dx) (f(x)= na_{n}x^{n-1}+ (n-1)a_{n-1}x^{n-2}+(n-2)a_{n-2}x^{n-3}+..............+2a_{2}x+a_{1}

**Some standard Derivatives:**

- (d/dx) (x
^{n})= nx^{n-1} - (d/dx) (sin x) = cos x
- (d/dx) (cos x) = - sin x
- (d/dx) (tanx) = sec
^{2}x - (d/dx) (cotx) = -cosec
^{2}x - (d/dx) (sec x) = sec x.tanx
- (d/dx) (sec x) = -cosec x.cot x
- (d/dx) (a
^{x})= a^{x}log_{e}a - (d/dx) (e
^{x})= e^{x} - (d/dx) (log
_{e}x) = 1/x

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

The link for NCERT textbook pdf is given below:

URL: ncert.nic.in/ncerts/l/kemh113.pdf

NCERT Class 11 Maths chapter 13 notes will be very much helpful for students to score maximum marks in their 11 board exams. In Limits and Derivatives Class 11 chapter 13 notes we have discussed many 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. NCERT Class 11 Maths chapter 13 is also very useful to cover major topics of Class 11 CBSE Maths Syllabus.

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

NCERT Class 11 Maths Chapter 13 Notes |

- NCERT Exemplar Class 11 Solutions
- NCERT Exemplar Class 11 Maths
- NCERT Exemplar Class 11 Physics
- NCERT Exemplar Class 11 Chemistry
- NCERT Exemplar Class 11 Biology

- NCERT Solutions for Class 11 Mathematics
- NCERT Solutions for Class 11 Chemistry
- NCERT Solutions for Class 11 Physics

1. Can we get to learn about negative limits in Limits and Derivatives Class 11 note?

Yes, we can get to learn about negative limits in the chapter. Limits can exist in the range from - infinite to + infinite.

2. How are limits important in Class 11 Maths chapter 13 notes?

Limits help us to determine the closest values of the function which is a part of the calculus.

3. Who were the mathematicians who invented Limits and derivatives NCERT Class 11 Math chapter 13?

The calculus part that is limits, derivatives, integrals, applications of integrals, continuity, infinite series were developed by Isaac Newton and Gottfried Wilhelm Leibniz.

4. Who is the father of calculus as in ncert notes for Class 11 Maths chapter 13?

Isaac Newton is the father of calculus.

5. What is the application of limits and derivatives in real life according to Class 11 Limits and Derivatives notes?

To calculate profit and loss, to locate the movement of the object continuously, it can also measure the values.

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