# NCERT solutions for Mathematics Exemplar Class 11 chapter 13 - Limits and Derivatives [Latest edition]

## Chapter 13: Limits and Derivatives

Solved ExamplesExercise
Solved Examples [Pages 227 - 239]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 13 Limits and Derivatives Solved Examples [Pages 227 - 239]

Solved Examples | Q 1 | Page 227

Evaluate lim_(x -> 2) 1/(x - 2) - (2(2x - 3))/(x^3 - 3x^2 + 2x)

Solved Examples | Q 2 | Page 228

Evaluate lim_(x -> 0) (sqrt(2 + x) - sqrt(2))/x

Solved Examples | Q 3 | Page 228

Find the positive integer n so that lim_(x -> 3) (x^n - 3^n)/(x - 3) = 108.

Solved Examples | Q 4 | Page 228

Evaluate lim_(x -> pi/2) (secx - tanx)

Solved Examples | Q 5 | Page 229

Evaluate lim_(x -> 0)  (sin(2 + x) - sin(2 - x))/x

Solved Examples | Q 6 | Page 229

Find the derivative of f(x) = ax + b, where a and b are non-zero constants, by first principle

Solved Examples | Q 7 | Page 229

Find the derivative of f(x) = ax2 + bx + c, where a, b and c are none-zero constant, by first principle.

Solved Examples | Q 8 | Page 230

Find the derivative of f(x) = x3, by first principle.

Solved Examples | Q 9 | Page 230

Find the derivative of f(x) = 1/x by first principle.

Solved Examples | Q 10 | Page 230

Find the derivative of f(x) = sin x, by first principle.

Solved Examples | Q 11 | Page 231

Find the derivative of f(x) = xn, where n is positive integer, by first principle.

Solved Examples | Q 12 | Page 231

Find the derivative of 2x4 + x.

Solved Examples | Q 13 | Page 232

Find the derivative of x2 cosx.

Solved Examples | Q 14 | Page 232

Evaluate lim_(x -> pi/6) (2sin^2x + sin x - 1)/(2sin^2 x - 3sin x + 1)

Solved Examples | Q 15 | Page 233

Evaluate lim_(x -> 0) (tanx - sinx)/(sin^3x)

Solved Examples | Q 16 | Page 233

Evaluate lim_(x -> a) (sqrt(a + 2x) - sqrt(3x))/(sqrt(3a + x) - 2sqrt(x))

Solved Examples | Q 17 | Page 234

Evaluate lim_(x -> 0) (cos ax - cos bx)/(cos cx - 1)

Solved Examples | Q 18 | Page 234

Evaluate lim_(h -> 0) ((a + h)^2 sin (a + h) - a^2 sina)/h

Solved Examples | Q 19 | Page 235

Find the derivative of f(x) = tan(ax + b), by first principle.

Solved Examples | Q 20 | Page 235

Find the derivative of f(x) = sqrt(sinx), by first principle.

Solved Examples | Q 21 | Page 236

Find the derivative of cosx/(1 + sinx)

#### Objective Type Questions from 22 to 28

Solved Examples | Q 22 | Page 237

lim_(x -> 0) sinx/(x(1 + cos x)) is equal to ______.

• 0

• 1/2

• 1

• –1

Solved Examples | Q 23 | Page 237

lim_(x -> pi/2) (1 - sin x)/cosx is equal to ______.

• 0

• –1

• 1

• Does not exit

Solved Examples | Q 24 | Page 238

lim_(x -> 0) |x|/x is equal to ______.

• 1

• –1

• 0

• Does not exists

Solved Examples | Q 25 | Page 238

lim_(x -> 1) [x - 1], where [.] is greatest integer function, is equal to ______.

• 1

• 2

• 0

• Does not exists

Solved Examples | Q 26 | Page 238

lim_(x -> 0) x sin  1/x is equal to ______.

• 0

• 1

• 1/2

• does not exist

Solved Examples | Q 27 | Page 239

lim_(n -> oo) (1 + 2 + 3 + ... + n)/n^2, n ∈ N, is equal to ______.

• 0

• 1

• 1/2

• 1/4

Solved Examples | Q 28 | Page 239

If f(x) = x sinx, then f" pi/2 is equal to ______.

• 0

• 1

• –1

• 1/2

Exercise [Pages 239 - 245]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 13 Limits and Derivatives Exercise [Pages 239 - 245]

Exercise | Q 1 | Page 239

Evaluate: lim_(x -> 3) (x^2 - 9)/(x - 3)

Exercise | Q 2 | Page 239

Evaluate: lim_(x -> 1/2) (4x^2 - 1)/(2x  - 1)

Exercise | Q 3 | Page 239

Evaluate: lim_(h -> 0) (sqrt(x + h) - sqrt(x))/h

Exercise | Q 4 | Page 239

Evaluate: lim_(x -> 0) ((x + 2)^(1/3) - 2^(1/3))/x

Exercise | Q 5 | Page 239

Evaluate: lim_(x -> 1) ((1 + x)^6 - 1)/((1 + x)^2 - 1)

Exercise | Q 6 | Page 239

Evaluate: lim_(x -> a) ((2 + x)^(5/2) - (a + 2)^(5/2))/(x - a)

Exercise | Q 7 | Page 240

Evaluate: lim_(x -> 1) (x^4 - sqrt(x))/(sqrt(x) - 1)

Exercise | Q 8 | Page 240

Evaluate: lim_(x -> 2) (x^2 - 4)/(sqrt(3x - 2) - sqrt(x + 2))

Exercise | Q 9 | Page 240

Evaluate: lim_(x -> sqrt(2)) (x^4 - 4)/(x^2 + 3sqrt(2x) - 8)

Exercise | Q 10 | Page 240

Evaluate: lim_(x -> 1) (x^7 - 2x^5 + 1)/(x^3 - 3x^2 + 2)

Exercise | Q 11 | Page 240

Evaluate: lim_(x -> 0) (sqrt(1 + x^3) - sqrt(1 - x^3))/x^2

Exercise | Q 12 | Page 240

Evaluate: lim_(x -> 3) (x^3 + 27)/(x^5 + 243)

Exercise | Q 13 | Page 240

Evaluate: lim_(x -> 1/2) (8x - 3)/(2x - 1) - (4x^2 + 1)/(4x^2 - 1)

Exercise | Q 14 | Page 240

Find ‘n’ if lim_(x -> 2) (x^n - 2^n)/(x - 2) = 80, x ∈ N

Exercise | Q 15 | Page 240

Evaluate: lim_(x -> 0) (sin 3x)/(sin 7x)

Exercise | Q 16 | Page 240

Evaluate: lim_(x -> 0) (sin^2 2x)/(sin^2 4x)

Exercise | Q 17 | Page 240

Evaluate: lim_(x -> 0) (1 - cos 2x)/x^2

Exercise | Q 18 | Page 240

Evaluate: lim_(x -> 0) (2 sin x - sin 2x)/x^3

Exercise | Q 19 | Page 240

Evaluate: lim_(x -> 0) (1 - cos mx)/(1 - cos nx)

Exercise | Q 20 | Page 240

Evaluate: lim_(x -> pi/3) (sqrt(1 - cos 6x))/(sqrt(2)(pi/3 - x))

Exercise | Q 21 | Page 240

Evaluate: lim_(x -> pi/4)  (sin x - cosx)/(x - pi/4)

Exercise | Q 22 | Page 240

Evaluate: lim_(x -> pi/6) (sqrt(3) sin x - cos x)/(x - pi/6)

Exercise | Q 22 | Page 240

Evaluate: lim_(x -> pi/6) (sqrt(3) sin x - cos x)/(x - pi/6)

Exercise | Q 23 | Page 240

Evaluate: lim_(x -> 0) (sin 2x + 3x)/(2x + tan 3x)

Exercise | Q 24 | Page 240

Evaluate: lim_(x -> a) (sin x - sin a)/(sqrt(x) - sqrt(a))

Exercise | Q 25 | Page 240

Evaluate: lim_(x -> pi/6) (cot^2 x - 3)/("cosec"  x - 2)

Exercise | Q 26 | Page 240

Evaluate: lim_(x -> 0) (sqrt(2) - sqrt(1 + cos x))/(sin^2x)

Exercise | Q 27 | Page 240

Evaluate: lim_(x -> 0) (sin x - 2 sin 3x + sin 5x)/x

Exercise | Q 28 | Page 240

If lim_(x -> 1) (x^4 - 1)/(x - 1) = lim_(x -> k) (x^3 - l^3)/(x^2 - k^2), then find the value of k.

#### Differentiate the functions w. r. to x in 29 to 42

Exercise | Q 29 | Page 240

(x^4 + x^3 + x^2 + 1)/x

Exercise | Q 30 | Page 240

(x + 1/x)^3

Exercise | Q 31 | Page 240

(3x + 5)(1 + tan x)

Exercise | Q 32 | Page 241

(sec x – 1)(sec x + 1)

Exercise | Q 33 | Page 241

(3x + 4)/(5x^2 - 7x + 9)

Exercise | Q 34 | Page 241

(x^5 - cosx)/sinx

Exercise | Q 35 | Page 241

(x^2 cos  pi/4)/sinx

Exercise | Q 36 | Page 241

(ax2 + cot x)(p + q cos x)

Exercise | Q 37 | Page 241

(a + b sin x)/(c + d cos x)

Exercise | Q 38 | Page 241

(sin x + cos x)2

Exercise | Q 39 | Page 241

(2x – 7)2 (3x + 5)3

Exercise | Q 40 | Page 241

x2 sin x + cos 2x

Exercise | Q 41 | Page 241

sin3x cos3x

Exercise | Q 42 | Page 241

1/(ax^2 + bx + c)

#### Long Answer: Differentiate the functions with respect to ‘x’ in 43 to 46 using first principle.

Exercise | Q 43 | Page 241

cos (x2 + 1)

Exercise | Q 44 | Page 241

(ax + b)/(cx + d)

Exercise | Q 45 | Page 241

x^(2/3)

Exercise | Q 46 | Page 241

x cos x

#### Evaluate the following limits in 47 to 53.

Exercise | Q 47 | Page 241

lim_(y -> 0) ((x + y) sec(x + y) - x sec x)/y

Exercise | Q 48 | Page 241

lim_(x -> 0) ((sin(alpha + beta) x + sin(alpha - beta)x + sin 2alpha x))/(cos 2betax - cos 2alphax) * x

Exercise | Q 49 | Page 241

lim_(x -> pi/4) (tan^3x - tan x)/(cos(x + pi/4))

Exercise | Q 50 | Page 241

lim_(x -> pi) (1 - sin  x/2)/(cos  x/2 (cos  x/4 - sin  x/4))

Exercise | Q 51 | Page 241

Show that lim_(x -> 4) |x - 4|/(x - 4) does not exists

Exercise | Q 52 | Page 242

Let f(x) = {{:((k cos x)/(pi - 2x)",", "when"  x ≠ pi/2),(3",", x = pi/2  "and if"  f(x) = f(pi/2)):} find the value of k.

Exercise | Q 53 | Page 242

If f(x) = {{:(x + 2",",  x ≤ - 1),(cx^2",", x > -1):}, find 'c' if lim_(x -> -1) f(x) exists

#### Objective Type Questions from 54 to 76

Exercise | Q 54 | Page 242

lim_(x -> pi) sinx/(x - pi) is equal to ______.

• 1

• 2

• – 1

• – 2

Exercise | Q 55 | Page 242

lim_(x -> 0) (x^2 cosx)/(1 - cosx) is ______.

• 2

• 3/2

• (-3)/2

• 1

Exercise | Q 56 | Page 242

lim_(x -> 0) ((1 + x)^n - 1)/x is equal to ______.

• n

• 1

• – n

• 0

Exercise | Q 57 | Page 242

lim_(x -> 1) (x^m - 1)/(x^n - 1) is ______.

• 1

• m/n

• - m/n

• m^2/n^2

Exercise | Q 58 | Page 242

lim_(x -> 0) (1 - cos 4theta)/(1 - cos 6theta) is ______.

• 4/9

• 1/2

• (-1)/2

• –1

Exercise | Q 59 | Page 243

lim_(x -> 0) ("cosec" x - cot x)/x is equal to ______.

• -1/2

• 1

• 1/2

• – 1

Exercise | Q 60 | Page 243

lim_(x -> 0) sinx/(sqrt(x + 1) - sqrt(1 - x) is ______.

• 2

• 0

• 1

• –1

Exercise | Q 61 | Page 243

lim_(x -> pi/4) (sec^2x - 2)/(tan x - 1) is equal to ______.

• 3

• 1

• 0

• 2

Exercise | Q 62 | Page 243

lim_(x -> 1) ((sqrt(x) - 1)(2x - 3))/(2x^2 + x - 3) is ______.

• 1/10

• (-1)/10

• 1

• None of these

Exercise | Q 63 | Page 243

If f(x) = {{:(sin[x]/[x]",", [x] ≠ 0),(0",", [x] = 0):}, where [.] denotes the greatest integer function, then lim_(x -> 0) f(x) is equal to ______.

• 1

• 0

• – 1

• None of these

Exercise | Q 64 | Page 243

lim_(x -> 0) |sinx|/x is ______.

• 1

• –1

• does not exist

• None of these

Exercise | Q 65 | Page 243

If f(x) = {{:(x^2 - 1",", 0 < x < 2),(2x + 3",", 2 ≤ x < 3):}, the quadratic equation whose roots are lim_(x -> 2^-) f(x) and lim_(x -> 2^+) f(x) is ______.

• x2 – 6x + 9 = 0

• x2 – 7x + 8 = 0

• x2 – 14x + 49 = 0

• x2 – 10x + 21 = 0

Exercise | Q 66 | Page 244

lim_(x -> 0) (tan 2x - x)/(3x - sin x) is equal to ______.

• 2

• 1/2

• -1/2

• 1/4

Exercise | Q 67 | Page 244

Let f(x) = x – [x]; ∈ R, then f'(1/2) is ______.

• 3/2

• 1

• 0

• –1

Exercise | Q 68 | Page 244

If y = sqrt(x) + 1/sqrt(x), then(dy)/(dx) at x = 1 is  ______.

• 1

• 1/2

• 1/sqrt(2)

• 0

Exercise | Q 69 | Page 244

if f(x) = (x - 4)/(2sqrt(x)), then f'(1) is ______.

• 5/4

• 4/5

• 1

• 0

Exercise | Q 70 | Page 244

If y = (1 + 1/x^2)/(1 - 1/x^2) then (dy)/(dx) is ______.

• (-4x)/(x^2 - 1)^2

• (-4x)/(x^2 - 1)

• (1 - x^2)/(4x)

• (4x)/(x^2 - 1)

Exercise | Q 71 | Page 244

If y = (sin x + cos x)/(sin x - cos x), then (dy)/(dx) at x = 0 is ______.

• –2

• 0

• 1/2

• Does not exist

Exercise | Q 72 | Page 245

If y = (sin(x + 9))/cosx then (dy)/(dx) at x = 0 is ______.

• cos 9

• sin 9

• 0

• 1

Exercise | Q 73 | Page 245

If f(x) = 1 + x + x^2/2 + ... + x^100/100, then f'(1) is equal to ______.

• 1/100

• 100

• does not exist

• 0

Exercise | Q 74 | Page 244

If f(x) = (x^n - a^n)/(x - a) for some constant, a, then f'(a) is equal to ______.

• 1

• 0

• does not exist

• 1/2

Exercise | Q 75 | Page 245

If f(x) = x^100 + x^99 .... +  x + 1, then f'(1) is equal to ______.

• 5050

• 5049

• 5051

• 50051

Exercise | Q 76 | Page 245

If f(x) = 1 - x + x^2 - x^3 + ... -x^99 + x^100, then f'(1) is equal to ______.

• 150

• – 50

• – 150

• 50

#### Fill in the blanks in 77 to 80

Exercise | Q 77 | Page 245

If f(x) = tanx/(x - pi), then lim_(x -> pi) f(x) = ______.

Exercise | Q 78 | Page 245

lim_(x -> 0) (sin mx cot  x/sqrt(3)) = 2, then m = ______.

Exercise | Q 79 | Page 245

If y = 1 + x/(1!) + x^2/(2!) + x^3/(3!) + ..., then (dy)/(dx) = ______.

Exercise | Q 80 | Page 245

lim_(x -> 3^+) x/([x]) = ______.

## Chapter 13: Limits and Derivatives

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 11 chapter 13 - Limits and Derivatives

NCERT solutions for Mathematics Exemplar Class 11 chapter 13 (Limits and Derivatives) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 11 chapter 13 Limits and Derivatives are Limits of Exponential Functions, Derivative of Slope of Tangent of the Curve, Theorem for Any Positive Integer n, Graphical Interpretation of Derivative, Derive Derivation of x^n, Algebra of Derivative of Functions, Derivative of Polynomials and Trigonometric Functions, Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically, Limits of Logarithmic Functions, Intuitive Idea of Derivatives, Introduction of Limits, Introduction to Calculus, Algebra of Limits, Limits of Polynomials and Rational Functions, Introduction of Derivatives, Limits of Trigonometric Functions.

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