#### Online Mock Tests

#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 13: Limits and Derivatives

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

#### Short Answer

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

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

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

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

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

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

Find the derivative of f(x) = ax^{2} + bx + c, where a, b and c are none-zero constant, by first principle.

Find the derivative of f(x) = x^{3}, by first principle.

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

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

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

Find the derivative of 2x^{4} + x.

Find the derivative of x^{2} cosx.

#### Long Answer

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

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

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

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

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

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

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

Find the derivative of `cosx/(1 + sinx)`

#### Objective Type Questions from 22 to 28

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

0

`1/2`

1

–1

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

0

–1

1

Does not exit

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

1

–1

0

Does not exists

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

1

2

0

Does not exists

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

0

1

`1/2`

does not exist

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

0

1

`1/2`

`1/4`

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

0

1

–1

`1/2`

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

#### Short Answer

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

`(x + 1/x)^3`

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

(sec x – 1)(sec x + 1)

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

`(x^5 - cosx)/sinx`

`(x^2 cos pi/4)/sinx`

(ax^{2} + cot x)(p + q cos x)

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

(sin x + cos x)^{2}

(2x – 7)^{2} (3x + 5)^{3}

x^{2} sin x + cos 2x

sin^{3}x cos^{3}x

`1/(ax^2 + bx + c)`

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

cos (x^{2} + 1)

`(ax + b)/(cx + d)`

`x^(2/3)`

x cos x

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

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

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

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

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

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

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.

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

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

1

2

– 1

– 2

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

2

`3/2`

`(-3)/2`

1

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

n

1

– n

0

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

1

`m/n`

`- m/n`

`m^2/n^2`

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

`4/9`

`1/2`

`(-1)/2`

–1

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

`-1/2`

1

`1/2`

– 1

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

2

0

1

–1

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

3

1

0

2

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

`1/10`

`(-1)/10`

1

None of these

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

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

1

–1

does not exist

None of these

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 ______.

x

^{2}– 6x + 9 = 0x

^{2}– 7x + 8 = 0x

^{2}– 14x + 49 = 0x

^{2}– 10x + 21 = 0

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

2

`1/2`

`-1/2`

`1/4`

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

`3/2`

1

0

–1

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

1

`1/2`

`1/sqrt(2)`

0

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

`5/4`

`4/5`

1

0

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)`

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

–2

0

`1/2`

Does not exist

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

cos 9

sin 9

0

1

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

`1/100`

100

does not exist

0

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`

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

5050

5049

5051

50051

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

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

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

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

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

## Chapter 13: Limits and Derivatives

## 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.

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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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