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 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.

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

NCERT CLASS 11 CHAPTER 13 NOTES

Limits:

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:

Algebra of Limits:

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)

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) 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 .

Derivatives:

1. 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

2. If f is real valued function then limits exists for

f(x) = lim_h→0 (f(x+h)-f(x))/h

Algebra of Derivative of Functions:

1. 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)

2. 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. 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. 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

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 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) (xⁿ)= nx^n-1
2. (d/dx) (sin x) = cos x
3. (d/dx) (cos x) = - sin x
4. (d/dx) (tanx) = sec²x
5. (d/dx) (cotx) = -cosec²x
6. (d/dx) (sec x) = sec x.tanx
7. (d/dx) (sec x) = -cosec x.cot x
8. (d/dx) (a^x)= a^xlog_ea
9. (d/dx) (e^x)= e^x
10. (d/dx) (log_ex) = 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

Significance of NCERT Class 11 Maths Chapter 13 Notes:

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.

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