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

## Mathematics Textbook for Class 11

#### NCERT Mathematics Class 11 ## Chapter 13: Limits and Derivatives

#### Chapter 13: Limits and Derivatives solutions [Pages 301 - 303]

Q 1 | Page 301

Evaluate the Given limit lim_(x -> 3) x + 3

Q 2 | Page 301

Evaluate the Given limit: lim_(x -> pi) (x - 22/7)

Q 3 | Page 301

Evaluate the Given limit: lim_(r -> l) pir^2

Q 4 | Page 301

Evaluate the Given limit: lim_(x -> 4) (4x + 3)/(x - 2)

Q 5 | Page 301

Evaluate the Given limit: lim_(x-> -1) (x^10 + x^5 +1)/(x -1)

Q 6 | Page 301

Evaluate the Given limit: lim_(x -> 0) ((x+1)^5 - 1)/x

Q 7 | Page 301

Evaluate the Given limit: lim_(x- > 2) (3x^2 - x - 10)/(x^2- 4)

Q 8 | Page 301

Evaluate the Given limit lim_(x -> 3) (x^4 - 81)/(2x^2-5x - 3)

Q 9 | Page 301

Evaluate the Given limit: lim_(x-> 0) (ax + b)/(cx + 1)

Q 10 | Page 301

Evaluate the Given limit: lim_(z -> 1) (z^(1/3) - 1)/(z^(1/6) -1)

Q 11 | Page 301

Evaluate the Given limit   lim_(x -> 1) (ax^2 + bx  + c)/(cx^2 + bx + a), a+b+c != 0

Q 12 | Page 301

Evaluate the Given limit: lim_(x -> -2) (1/x + 1/2)/(x + 2)

Q 13 | Page 301

Evaluate the Given limit lim_(x -> 0) (sin ax)/ (bx)

Q 14 | Page 301

Evaluate the Given limit: lim_(x -> 0) (sin ax)/(sin bx), a, b != 0

Q 15 | Page 302

Evaluate the Given limit lim_(x -> pi) (sin(pi - x))/(pi (pi - x))

Q 16 | Page 302

Evaluate the given limit lim_(x ->0) cos x/(pi - x)

Q 17 | Page 302

Evaluate the Given limit: lim_(x -> 0) (cos 2x -1)/(cos x - 1)

Q 18 | Page 302

Evaluate the Given limit lim_(x -> 0) (ax +  xcos x)/(b sin x)

Q 19 | Page 302

Evaluate the Given limit lim_(x -> 0) sec x

Q 20 | Page 302

Evaluate the Given limit lim_(x -> 0) (sin ax + bx)/(ax + sin bx) a, b, a+ b != 0

Q 21 | Page 302

Evaluate the Given limit lim_(x -> 0) (cosec x -  cot x)

Q 22 | Page 302

Evaluate the Given limit lim_(x -> (pi/2))     (tan 2x)/(x - pi/2)

Q 23 | Page 302

Find  lim_(x -> 0) f(x) and lim_(x -> 1) f(x) where f(x) = {(2x + 3, x <= 0),(3(x+1), x > 0):}

Q 24 | Page 302

Find  lim_(x -> 1) f(x) where f(x) = {(x^2 -1, x < = 1), (-x^2 -1, x > 1):}

Q 25 | Page 302

Evaluate lim_(x -> 0) f(x) where f(x) = { (|x|/x, x != 0),(0, x = 0):}

Q 26 | Page 302

Find lim_(x -> 0) f(x) where f(x) = {(x/|x|, x != 0),(0, x = 0):}

Q 27 | Page 302

Find lim_(x -> 5) f(x), where f(x)  = |x| - 5

Q 28 | Page 302

Suppose f(x)  = {(a+bx, x < 1),(4, x = 1),(b-ax, x > 1):}  and if lim_(x -> 1) f(x) = f(1) what are possible values of and b?

Q 29 | Page 303

Let a1, a2, . . ., an be fixed real numbers and define a function f ( x) = ( x − a1 ) ( x − a2 )...( x − an ) .

What is lim_(x -> a_1) f(x) ? For some a ≠ a1, a2, ..., an, compute lim_(x -> a) f(x)

Q 30 | Page 303

If f(x)  = {(|x| +  1,x < 0), (0, x = 0),(|x| -1, x > 0):}

For what value (s) of a does lim_(x -> a)  f(x) exists?

Q 31 | Page 303

If the function f(x) satisfies lim_(x -> 1) (f(x) - 2)/(x^2 - 1) = pi, evaluate lim_(x -> 1) f(x)

Q 32 | Page 303

if f(x) = { (mx^2 + n, x < 0),(nx + m, 0<= x <= 1),(nm^3 + m, x > 1):}

For what integers m and n does lim_(x-> 0) f(x) and lim_(x -> 1) f(x) exist?

#### Chapter 13: Limits and Derivatives solutions [Pages 312 - 313]

Q 1 | Page 312

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

Q 2 | Page 312

Find the derivative of 99x at x = 100.

Q 3 | Page 312

Find the derivative of at = 1.

Q 4.1 | Page 312

Find the derivative of the following functions from first principle.

x3 – 27

Q 4.2 | Page 312

Find the derivative of the following functions from first principle.

(x – 1) (– 2)

Q 4.3 | Page 312

Find the derivative of the following functions from first principle.

1/x^2

Q 4.4 | Page 312

Find the derivative of the following functions from first principle.

(x+1)/(x -1)

Q 5 | Page 312

For the function

f(x) = x^100/100 + x^99/99 + ...+ x^2/2 + x + 1

Prove that f'(1) = 100 f'(0)

Q 6 | Page 313

Find the derivative of x^n  + ax^(n-1) + a^2 x^(n-2) + ...+ a^(n -1) x + a^n for some fixed real number a

Q 7.1 | Page 313

For some constants a and b, find the derivative of (– a) (x – b)

Q 7.2 | Page 313

For some constants a and b, find the derivative of (ax2 + b)2

Q 7.3 | Page 313

For some constants a and b, find the derivative of

(x-a)/(x -b)

Q 8 | Page 313

Find the derivative of (x^n - a^n)/(x -a) for some constant a.

Q 9.1 | Page 313

Find the derivative of 2x - 3/4

Q 9.2 | Page 313

Find the derivative of (5x3 + 3– 1) (x – 1)

Q 9.3 | Page 313

Find the derivative of x–3 (5 + 3x)

Q 9.4 | Page 313

Find the derivative of x5 (3 – 6x–9)

Q 9.5 | Page 313

Find the derivative of x–4 (3 – 4x–5)

Q 9.6 | Page 313

Find the derivative of 2/(x + 1) - x^2/(3x -1)

Q 10 | Page 313

Find the derivative of cos x from first principle.

Q 11.1 | Page 313

Find the derivative of the following functions

sin x cos x

Q 11.2 | Page 313

Find the derivative of the following functions: sec x

Q 11.3 | Page 313

Find the derivative of the following functions:

5 sec x + 4 cos x

Q 11.4 | Page 313

Find the derivative of the following functions:

cosec x

Q 11.5 | Page 313

Find the derivative of the following functions:

3cot x + 5cosec x

Q 11.6 | Page 313

Find the derivative of the following functions: 5sin x – 6cos x + 7

Q 11.7 | Page 313

Find the derivative of the following functions:

2tan x – 7sec x

#### Chapter 13: Limits and Derivatives solutions [Pages 317 - 318]

Q 1.1 | Page 317

Find the derivative of the following functions from first principle:

−x

Q 1.2 | Page 317

Find the derivative of the following functions from first principle:

(–x)–1

Q 1.3 | Page 317

Find the derivative of the following functions from first principle:

sin (x + 1)

Q 1.4 | Page 317

Find the derivative of the following functions from first principle:

cos (x - pi/8)

Q 2 | Page 317

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): (x + a)

Q 3 | Page 317

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): (px+ q) (r/s + s)

Q 4 | Page 317

Find the derivative of the following functions (it is to be understood that abcdp, q, rand s are fixed non-zero constants and m and n are integers): (ax + b) (cx d)2

Q 5 | Page 317

Find the derivative of the following functions (it is to be understood that abcdp, q, rand s are fixed non-zero constants and m and n are integers): (ax + b)/(cx + d)

Q 6 | Page 317

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): (1 + 1/x)/(1- 1/x)

Q 7 | Page 317

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):  1/(ax^2 + bx + c)

Q 8 | Page 317

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) : (ax + b)/(px^2 + qx + r)

Q 9 | Page 317

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): (px^2 +qx + r)/(ax +b)

Q 10 | Page 317

Find the derivative of the following functions (it is to be understood that abcdp, q, rand s are fixed non-zero constants and m and n are integers) : a/x^4 = b/x^2 + cos x

Q 11 | Page 317

Find the derivative of the following functions (it is to be understood that abcdp, q, rand s are fixed non-zero constants and m and n are integers):  4sqrtx - 2

Q 12 | Page 317

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers): (ax + b)n

Q 13 | Page 317

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): (ax + b)n (cx + d)m

Q 14 | Page 317

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): sin (x + a)

Q 15 | Page 317

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): cosec x cot x

Q 16 | Page 317

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): cos x/(1 +  sin x)

Q 17 | Page 318

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) : (sin x + cos x)/(sin x - cos x)

Q 18 | Page 318

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) :  (sec x - 1)/(sec x + 1)

Q 19 | Page 318

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): sinn x

Q 20 | Page 318

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): (a + bsin x)/(c + dcosx)

Q 21 | Page 318

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): (sin(x + a))/ cos x

Q 22 | Page 318

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): x4 (5 sin x – 3 cos x)

Q 23 | Page 318

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): (x2 + 1) cos x

Q 24 | Page 318

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): (ax2 + sin x) (p + q cos x)

Q 25 | Page 318

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): (x + cos x)(x - tan x)

Q 26 | Page 318

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): (4x + 5sin x)/(3x + 7cos x)

Q 27 | Page 318

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):  (x^2 cos (pi/4))/sin x

Q 28 | Page 318

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): x/(1 + tan x)

Q 29 | Page 318

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): (x + sec x) (x – tan x)

Q 30 | Page 318

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): x/(sin^n x)

## Chapter 13: Limits and Derivatives

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

NCERT solutions for Class 11 Maths 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 Textbook for Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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

Using NCERT Class 11 solutions Limits and Derivatives exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer NCERT Textbook Solutions to score more in exam.

Get the free view of chapter 13 Limits and Derivatives Class 11 extra questions for Maths and can use Shaalaa.com to keep it handy for your exam preparation

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