NCERT Solution Class 11 Chapter 13- Limits and Derivatives

The NCERT Solutions For Class 11 Maths Chapter 13 Limits and Derivatives are arranged topic wise in a systematic manner. The NCERT Solutions are authored by the most experienced educators in the teaching industry, writing the solutions for every problem in a simpler way. These solutions pdf comes handy for a Class 11 student to understand the idea of Limits and Derivatives.

Download PDF of NCERT Solutions for Class 11 Maths Chapter 13- Limits and Derivatives

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Exercise 13.1 Solutions 32 Questions

Exercise 13.2 Solutions 11 Questions

Miscellaneous Exercise On Chapter 13 Solutions 30 Questions

Access Answers of Maths NCERT Class 11 Chapter 13

Exercise 13.1 page no: 301

1. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 1

Solution:

Given

NCERT Solutions Mathematics Class 11 Chapter 13 - 2

Substituting x = 3, we get

= 3 + 3

= 6

2. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 3

Solution:

Given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 4

Substituting x = π, we get

NCERT Solutions Mathematics Class 11 Chapter 13 - 5= (π – 22 / 7)

3. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 6

Solution:

Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 7

Substituting r = 1, we get

NCERT Solutions Mathematics Class 11 Chapter 13 - 8= π(1)2

= π

4. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 9

Solution:

Given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 10

Substituting x = 4, we get

NCERT Solutions Mathematics Class 11 Chapter 13 - 11= [4(4) + 3] / (4 – 2)

= (16 + 3) / 2

= 19 / 2

5. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 12

Solution:

Given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 13

Substituting x = -1, we get

NCERT Solutions Mathematics Class 11 Chapter 13 - 14

= [(-1)10 + (-1)5 + 1] / (-1 – 1)

= (1 – 1 + 1) / – 2

= – 1 / 2

6. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 15

Solution:

Given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 16

= [(0 + 1)5 – 1] / 0

=0

Since, this limit is undefined

Substitute x + 1 = y, then x = y – 1

NCERT Solutions Mathematics Class 11 Chapter 13 - 17

7. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 18

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 19

8. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 20

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 21

9. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 22

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 23

= [a (0) + b] / c (0) + 1

= b / 1

= b

10. Evaluate the Given limit: NCERT Solutions Mathematics Class 11 Chapter 13 - 24

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 25

11. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 26

Solution:

Given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 27

Substituting x = 1

NCERT Solutions Mathematics Class 11 Chapter 13 - 28

= [a (1)2 + b (1) + c] / [c (1)2 + b (1) + a]

= (a + b + c) / (a + b + c)

Given

NCERT Solutions Mathematics Class 11 Chapter 13 - 29

= 1

12. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 30

Solution:

By substituting x = – 2, we get

NCERT Solutions Mathematics Class 11 Chapter 13 - 31

13. Evaluate the Given limit:NCERT Solutions Mathematics Class 11 Chapter 13 - 32

Solution:

Given NCERT Solutions Mathematics Class 11 Chapter 13 - 33

NCERT Solutions Mathematics Class 11 Chapter 13 - 34

14. Evaluate the given limit: NCERT Solutions Mathematics Class 11 Chapter 13 - 35

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 36

15. Evaluate the given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 37

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 38

16. Evaluate the given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 39

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 40

17. Evaluate the given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 41

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 42

NCERT Solutions Mathematics Class 11 Chapter 13 - 43

18. Evaluate the given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 44

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 45

19. Evaluate the given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 46

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 47

20. Evaluate the given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 48

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 49

NCERT Solutions Mathematics Class 11 Chapter 13 - 50

21. Evaluate the given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 51

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 52

NCERT Solutions Mathematics Class 11 Chapter 13 - 53

22. Evaluate the given limit:
NCERT Solutions Mathematics Class 11 Chapter 13 - 54

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 55

NCERT Solutions Mathematics Class 11 Chapter 13 - 56

23.
NCERT Solutions Mathematics Class 11 Chapter 13 - 57

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 58

NCERT Solutions Mathematics Class 11 Chapter 13 - 59

NCERT Solutions Mathematics Class 11 Chapter 13 - 60

24. Find
NCERT Solutions Mathematics Class 11 Chapter 13 - 61, where

NCERT Solutions Mathematics Class 11 Chapter 13 - 62

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 63

NCERT Solutions Mathematics Class 11 Chapter 13 - 64

25. Evaluate
NCERT Solutions Mathematics Class 11 Chapter 13 - 65, where f(x) =
NCERT Solutions Mathematics Class 11 Chapter 13 - 66

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 67

NCERT Solutions Mathematics Class 11 Chapter 13 - 68

NCERT Solutions Mathematics Class 11 Chapter 13 - 69

26. Find
NCERT Solutions Mathematics Class 11 Chapter 13 - 70, where f (x) =
NCERT Solutions Mathematics Class 11 Chapter 13 - 71

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 72

NCERT Solutions Mathematics Class 11 Chapter 13 - 73

27. Find
NCERT Solutions Mathematics Class 11 Chapter 13 - 74, where
NCERT Solutions Mathematics Class 11 Chapter 13 - 75

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 76

NCERT Solutions Mathematics Class 11 Chapter 13 - 77

28. Suppose
NCERT Solutions Mathematics Class 11 Chapter 13 - 78and if
NCERT Solutions Mathematics Class 11 Chapter 13 - 79what are possible values of a and b

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 80

NCERT Solutions Mathematics Class 11 Chapter 13 - 81

29. Let a1, a2,………an be fixed real numbers and define a function

f (x) = (x – a1) (x – a2) ……. (x – an).

What is
NCERT Solutions Mathematics Class 11 Chapter 13 - 82For some a ≠ a1, a2, ……. an, compute
NCERT Solutions Mathematics Class 11 Chapter 13 - 83

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 84

NCERT Solutions Mathematics Class 11 Chapter 13 - 85

30. If NCERT Solutions Mathematics Class 11 Chapter 13 - 86 For what value (s) of a does NCERT Solutions Mathematics Class 11 Chapter 13 - 87exists?

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 88

NCERT Solutions Mathematics Class 11 Chapter 13 - 89

NCERT Solutions Mathematics Class 11 Chapter 13 - 90

NCERT Solutions Mathematics Class 11 Chapter 13 - 91

31. If the function f(x) satisfies NCERT Solutions Mathematics Class 11 Chapter 13 - 92, evaluateNCERT Solutions Mathematics Class 11 Chapter 13 - 93

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 94

NCERT Solutions Mathematics Class 11 Chapter 13 - 95

32. If NCERT Solutions Mathematics Class 11 Chapter 13 - 96 For what integers m and n does both NCERT Solutions Mathematics Class 11 Chapter 13 - 97and NCERT Solutions Mathematics Class 11 Chapter 13 - 98exist?

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 99

NCERT Solutions Mathematics Class 11 Chapter 13 - 100

NCERT Solutions Mathematics Class 11 Chapter 13 - 101


Exercise 13.2 page no: 312

1. Find the derivative of x2– 2 at x = 10

Solution:

Let f (x) = x2 – 2

From first principle

NCERT Solutions Mathematics Class 11 Chapter 13 - 102


NCERT Solutions Mathematics Class 11 Chapter 13 - 103

2. Find the derivative of x at x = 1.

Solution:

Let f (x) = x

Then,

NCERT Solutions Mathematics Class 11 Chapter 13 - 104


NCERT Solutions Mathematics Class 11 Chapter 13 - 105

3. Find the derivative of 99x at x = l00.

Solution:

Let f (x) = 99x,

From first principle

NCERT Solutions Mathematics Class 11 Chapter 13 - 106

= 99

4. Find the derivative of the following functions from first principle

(i) x3 – 27

(ii) (x – 1) (x – 2)

(iii) 1 / x2

(iv) x + 1 / x – 1

Solution:

(i) Let f (x) = x3 – 27

From first principle

NCERT Solutions Mathematics Class 11 Chapter 13 - 107


NCERT Solutions Mathematics Class 11 Chapter 13 - 108

(ii) Let f (x) = (x – 1) (x – 2)

From first principle

NCERT Solutions Mathematics Class 11 Chapter 13 - 109

NCERT Solutions Mathematics Class 11 Chapter 13 - 110

(iii) Let f (x) = 1 / x2

From first principle, we get

NCERT Solutions Mathematics Class 11 Chapter 13 - 111

NCERT Solutions Mathematics Class 11 Chapter 13 - 112

(iv) Let f (x) = x + 1 / x – 1

From first principle, we get

NCERT Solutions Mathematics Class 11 Chapter 13 - 113

NCERT Solutions Mathematics Class 11 Chapter 13 - 114

5. For the function NCERT Solutions Mathematics Class 11 Chapter 13 - 115 .Prove that f’ (1) =100 f’ (0).

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 116

NCERT Solutions Mathematics Class 11 Chapter 13 - 117

6. Find the derivative of NCERT Solutions Mathematics Class 11 Chapter 13 - 118 for some fixed real number a.

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 119

7. For some constants a and b, find the derivative of
(i) (x − a) (x − b)

(ii) (ax2 + b)2

(iii) x – a / x – b

Solution:

(i) (x – a) (x – b)

NCERT Solutions Mathematics Class 11 Chapter 13 - 120

(ii) (ax2 + b)2

NCERT Solutions Mathematics Class 11 Chapter 13 - 121

= 4ax(ax2 + b)

(iii) x – a / x – b

NCERT Solutions Mathematics Class 11 Chapter 13 - 122

NCERT Solutions Mathematics Class 11 Chapter 13 - 123

NCERT Solutions Mathematics Class 11 Chapter 13 - 124

8. Find the derivative of NCERT Solutions Mathematics Class 11 Chapter 13 - 125 for some constant a.

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 126

9. Find the derivative of

(i) 2x – 3 / 4

(ii) (5x3 + 3x – 1) (x – 1)

(iii) x-3 (5 + 3x)

(iv) x5 (3 – 6x-9)

(v) x-4 (3 – 4x-5)

(vi) (2 / x + 1) – x2 / 3x – 1

Solution:

(i)

NCERT Solutions Mathematics Class 11 Chapter 13 - 127

(ii)

NCERT Solutions Mathematics Class 11 Chapter 13 - 128

(iii)

NCERT Solutions Mathematics Class 11 Chapter 13 - 129

NCERT Solutions Mathematics Class 11 Chapter 13 - 130

(iv)

NCERT Solutions Mathematics Class 11 Chapter 13 - 131

(v)

NCERT Solutions Mathematics Class 11 Chapter 13 - 132

NCERT Solutions Mathematics Class 11 Chapter 13 - 133

(vi)

NCERT Solutions Mathematics Class 11 Chapter 13 - 134

NCERT Solutions Mathematics Class 11 Chapter 13 - 135

10. Find the derivative of cos x from first principle

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 136

NCERT Solutions Mathematics Class 11 Chapter 13 - 137

NCERT Solutions Mathematics Class 11 Chapter 13 - 138

NCERT Solutions Mathematics Class 11 Chapter 13 - 139

11. Find the derivative of the following functions:

(i) sin x cos x

(ii) sec x

(iii) 5 sec x + 4 cos x

(iv) cosec x

(v) 3 cot x + 5 cosec x

(vi) 5 sin x – 6 cos x + 7

(vii) 2 tan x – 7 sec x

Solution:

(i) sin x cos x

NCERT Solutions Mathematics Class 11 Chapter 13 - 140

NCERT Solutions Mathematics Class 11 Chapter 13 - 141

(ii) sec x

NCERT Solutions Mathematics Class 11 Chapter 13 - 142

NCERT Solutions Mathematics Class 11 Chapter 13 - 143

NCERT Solutions Mathematics Class 11 Chapter 13 - 144

(iii) 5 sec x + 4 cos x

NCERT Solutions Mathematics Class 11 Chapter 13 - 145

(iv) cosec x

NCERT Solutions Mathematics Class 11 Chapter 13 - 146

NCERT Solutions Mathematics Class 11 Chapter 13 - 147

NCERT Solutions Mathematics Class 11 Chapter 13 - 148

(v) 3 cot x + 5 cosec x

NCERT Solutions Mathematics Class 11 Chapter 13 - 149

NCERT Solutions Mathematics Class 11 Chapter 13 - 150

NCERT Solutions Mathematics Class 11 Chapter 13 - 151

NCERT Solutions Mathematics Class 11 Chapter 13 - 152

NCERT Solutions Mathematics Class 11 Chapter 13 - 153

NCERT Solutions Mathematics Class 11 Chapter 13 - 154

NCERT Solutions Mathematics Class 11 Chapter 13 - 155

NCERT Solutions Mathematics Class 11 Chapter 13 - 156

(vi)5 sin x – 6 cos x + 7

NCERT Solutions Mathematics Class 11 Chapter 13 - 157

NCERT Solutions Mathematics Class 11 Chapter 13 - 158

NCERT Solutions Mathematics Class 11 Chapter 13 - 159

(vii) 2 tan x – 7 sec x

NCERT Solutions Mathematics Class 11 Chapter 13 - 160

NCERT Solutions Mathematics Class 11 Chapter 13 - 161

NCERT Solutions Mathematics Class 11 Chapter 13 - 162

NCERT Solutions Mathematics Class 11 Chapter 13 - 163


Miscellaneous exercise page no: 317

1. Find the derivative of the following functions from first principle:

(i) –x 

(ii) (–x)–1 

(iii) sin (x + 1)

(iv) NCERT Solutions Mathematics Class 11 Chapter 13 - 164

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 165

(ii) (-x)-1

NCERT Solutions Mathematics Class 11 Chapter 13 - 166

NCERT Solutions Mathematics Class 11 Chapter 13 - 167

NCERT Solutions Mathematics Class 11 Chapter 13 - 168

NCERT Solutions Mathematics Class 11 Chapter 13 - 169

NCERT Solutions Mathematics Class 11 Chapter 13 - 170

= 1 / x2

(iii) sin (x + 1)

NCERT Solutions Mathematics Class 11 Chapter 13 - 171

NCERT Solutions Mathematics Class 11 Chapter 13 - 172

NCERT Solutions Mathematics Class 11 Chapter 13 - 173

(iv) NCERT Solutions Mathematics Class 11 Chapter 13 - 174

NCERT Solutions Mathematics Class 11 Chapter 13 - 175

We get,

NCERT Solutions Mathematics Class 11 Chapter 13 - 176

NCERT Solutions Mathematics Class 11 Chapter 13 - 177

NCERT Solutions Mathematics Class 11 Chapter 13 - 178

NCERT Solutions Mathematics Class 11 Chapter 13 - 179

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers):

2. (x + a)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 180

3. (px + q) (r / x + s)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 181

NCERT Solutions Mathematics Class 11 Chapter 13 - 182

NCERT Solutions Mathematics Class 11 Chapter 13 - 183

NCERT Solutions Mathematics Class 11 Chapter 13 - 184

4. (ax + b) (cx d)2

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 185

NCERT Solutions Mathematics Class 11 Chapter 13 - 186

5. (ax + b) / (cx + d)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 187

NCERT Solutions Mathematics Class 11 Chapter 13 - 188

6. (1 + 1 / x) / (1 – 1 / x)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 189

NCERT Solutions Mathematics Class 11 Chapter 13 - 190

NCERT Solutions Mathematics Class 11 Chapter 13 - 191

NCERT Solutions Mathematics Class 11 Chapter 13 - 192

7. 1 / (ax2 + bx + c)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 193

NCERT Solutions Mathematics Class 11 Chapter 13 - 194

8. (ax + b) / px2 + qx + r

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 195

NCERT Solutions Mathematics Class 11 Chapter 13 - 196

NCERT Solutions Mathematics Class 11 Chapter 13 - 197

9. (px2 + qx + r) / ax + b

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 198

NCERT Solutions Mathematics Class 11 Chapter 13 - 199

10. (a / x4) – (b / x2) + cox x

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 200

NCERT Solutions Mathematics Class 11 Chapter 13 - 201

NCERT Solutions Mathematics Class 11 Chapter 13 - 202

11. NCERT Solutions Mathematics Class 11 Chapter 13 - 203

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 204

NCERT Solutions Mathematics Class 11 Chapter 13 - 205

12. (ax + b)n

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 206

NCERT Solutions Mathematics Class 11 Chapter 13 - 207

NCERT Solutions Mathematics Class 11 Chapter 13 - 208

NCERT Solutions Mathematics Class 11 Chapter 13 - 209

NCERT Solutions Mathematics Class 11 Chapter 13 - 210

NCERT Solutions Mathematics Class 11 Chapter 13 - 211

13. (ax + b)n (cx + d)m

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 212

NCERT Solutions Mathematics Class 11 Chapter 13 - 213

NCERT Solutions Mathematics Class 11 Chapter 13 - 214

NCERT Solutions Mathematics Class 11 Chapter 13 - 215

NCERT Solutions Mathematics Class 11 Chapter 13 - 216

14. sin (x + a)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 217

NCERT Solutions Mathematics Class 11 Chapter 13 - 218

NCERT Solutions Mathematics Class 11 Chapter 13 - 219

NCERT Solutions Mathematics Class 11 Chapter 13 - 220

15. cosec x cot x

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 221

NCERT Solutions Mathematics Class 11 Chapter 13 - 222

So, we get

NCERT Solutions Mathematics Class 11 Chapter 13 - 223

NCERT Solutions Mathematics Class 11 Chapter 13 - 224

NCERT Solutions Mathematics Class 11 Chapter 13 - 225

NCERT Solutions Mathematics Class 11 Chapter 13 - 226

16. NCERT Solutions Mathematics Class 11 Chapter 13 - 227

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 228

NCERT Solutions Mathematics Class 11 Chapter 13 - 229

17.
NCERT Solutions Mathematics Class 11 Chapter 13 - 230

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 231

NCERT Solutions Mathematics Class 11 Chapter 13 - 232

NCERT Solutions Mathematics Class 11 Chapter 13 - 233

18.
NCERT Solutions Mathematics Class 11 Chapter 13 - 234

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 235

NCERT Solutions Mathematics Class 11 Chapter 13 - 236

NCERT Solutions Mathematics Class 11 Chapter 13 - 237

19. sinn x

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 238

NCERT Solutions Mathematics Class 11 Chapter 13 - 239

NCERT Solutions Mathematics Class 11 Chapter 13 - 240

NCERT Solutions Mathematics Class 11 Chapter 13 - 241

NCERT Solutions Mathematics Class 11 Chapter 13 - 242

20. NCERT Solutions Mathematics Class 11 Chapter 13 - 243

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 244

NCERT Solutions Mathematics Class 11 Chapter 13 - 245

21.
NCERT Solutions Mathematics Class 11 Chapter 13 - 246

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 247

NCERT Solutions Mathematics Class 11 Chapter 13 - 248

NCERT Solutions Mathematics Class 11 Chapter 13 - 249

NCERT Solutions Mathematics Class 11 Chapter 13 - 250

22. x4 (5 sin x – 3 cos x)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 251

NCERT Solutions Mathematics Class 11 Chapter 13 - 252

23. (x2 + 1) cos x

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 253

24. (ax2 + sin x) (p + q cos x)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 254

25. NCERT Solutions Mathematics Class 11 Chapter 13 - 255

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 256

NCERT Solutions Mathematics Class 11 Chapter 13 - 257

NCERT Solutions Mathematics Class 11 Chapter 13 - 258

26. NCERT Solutions Mathematics Class 11 Chapter 13 - 259

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 260

NCERT Solutions Mathematics Class 11 Chapter 13 - 261

27. NCERT Solutions Mathematics Class 11 Chapter 13 - 262

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 263

28. NCERT Solutions Mathematics Class 11 Chapter 13 - 264

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 265

NCERT Solutions Mathematics Class 11 Chapter 13 - 266

NCERT Solutions Mathematics Class 11 Chapter 13 - 267

NCERT Solutions Mathematics Class 11 Chapter 13 - 268

NCERT Solutions Mathematics Class 11 Chapter 13 - 269

29. (x + sec x) (x – tan x)

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 270

NCERT Solutions Mathematics Class 11 Chapter 13 - 271

NCERT Solutions Mathematics Class 11 Chapter 13 - 272

NCERT Solutions Mathematics Class 11 Chapter 13 - 273


NCERT Solutions Mathematics Class 11 Chapter 13 - 274

NCERT Solutions Mathematics Class 11 Chapter 13 - 275

NCERT Solutions Mathematics Class 11 Chapter 13 - 276

NCERT Solutions Mathematics Class 11 Chapter 13 - 277

NCERT Solutions Mathematics Class 11 Chapter 13 - 278

NCERT Solutions Mathematics Class 11 Chapter 13 - 279

NCERT Solutions Mathematics Class 11 Chapter 13 - 280

NCERT Solutions Mathematics Class 11 Chapter 13 - 281

30. NCERT Solutions Mathematics Class 11 Chapter 13 - 282

Solution:

NCERT Solutions Mathematics Class 11 Chapter 13 - 283

NCERT Solutions Mathematics Class 11 Chapter 13 - 284


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

Class 11 Maths Chapter 13 Limits and Derivatives of NCERT Solutions include the following:

13.1 Introduction

This section introduces the branch of mathematics called calculus, derivative, limits and algebra of limits. Derivatives of certain standard functions are also discussed.

  • Calculus is used in electrical engineering to find the length of wire to be used between two stations which are miles apart.
  • Calculus is used to find the velocity and trajectory of an object.
  • Calculus is used in operations research to improve efficiency, productivity and earn profits.

13.2 Intuitive Idea of Derivatives

This section defines the derivative of a function by giving a real-life example of distance travelled at different intervals of time. 

The derivative is the rate of change at which one quantity varies concerning another. If you run a business (say selling candies), derivatives can help you decide what quantity you should sell. The data of your business performance over the past few months is to be taken and the trend needs to be analyzed by plotting a curve. Derivative defines the instantaneous slope (dy/dx) of this graph. 

The slope gives an idea of the marginal cost (change in cost/change in unit produced) of producing every additional unit at that instant. If this cost is greater than 10 bucks that is the selling price, then a loss is incurred at the point.

For better clarity, refer to the table given below.

For 0-300 units, if the cost > sales, then it’s a loss

For 400-600 units, if the sales > cost, then it’s a profit 

13.3 Limits

This section explains limits with an interesting example, left-hand limit and right-hand limit with solved problems.

If an ice cube is dropped into a glass of warm water and the temperature is measured with time, it is an example of limit as the time approaches infinity. The time measured here is the temperature slowly attaining the room temperature where the glass was stored.

13.3.1 Algebra of limits

This section demonstrates the output of sum, difference, product and quotient of limits. 

13.3.2 Limits of polynomials and rational functions

The limits of polynomials and rational functions are elaborated along with solved examples.

13.4 Limits of Trigonometric Functions

This section deals with the theorems and prepositions for functions, which are helpful in calculating the limits of some trigonometric functions.

13.5 Derivatives

This section helps the student to find solutions on different derivative problems.

Money related establishments need to anticipate the adjustments in the estimation of a specific stock knowing its present worth. In this, and numerous such cases, it is important to know how a specific parameter is changing in accordance with the other parameter. The heart of the text is derivative of a function at a given point in its domain of definition.

13.5.1 Algebra of derivatives of functions

This section demonstrates the output of sum, difference, product and quotient of derivatives.

13.5.2 Derivative of polynomials and trigonometric functions

This section deals with the theorems and problems for functions which help calculate the derivatives of some trigonometric functions.

A few points on Chapter 13 Limits and Derivatives

  • 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.
  • For a function f and a real number, a lim x → a f(x) and f (a) may not be the same.

The solutions given in the pdf contain a definite clarification for each problem. The NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives are accessible on your fingertips!! The solutions are set up so that a student will have a comprehension of the split-up marks for every part, which thus benefits them to identify the time and energy they need to allocate to every section. High scoring students have suggested NCERT solutions to get sound information on each topic.

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