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

## Chapter 13: Limits and Derivatives

Exercise 13.1Exercise 13.2Miscellaneous Exercise
Exercise 13.1 [Pages 301 - 303]

### NCERT solutions for Class 11 Mathematics Chapter 13 Limits and Derivatives Exercise 13.1 [Pages 301 - 303]

Exercise 13.1 | Q 1 | Page 301

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

Exercise 13.1 | Q 2 | Page 301

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

Exercise 13.1 | Q 3 | Page 301

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

Exercise 13.1 | Q 4 | Page 301

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

Exercise 13.1 | Q 5 | Page 301

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

Exercise 13.1 | Q 6 | Page 301

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

Exercise 13.1 | Q 7 | Page 301

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

Exercise 13.1 | Q 8 | Page 301

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

Exercise 13.1 | Q 9 | Page 301

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

Exercise 13.1 | Q 10 | Page 301

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

Exercise 13.1 | Q 11 | Page 301

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

Exercise 13.1 | Q 12 | Page 301

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

Exercise 13.1 | Q 13 | Page 301

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

Exercise 13.1 | Q 14 | Page 301

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

Exercise 13.1 | Q 15 | Page 302

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

Exercise 13.1 | Q 16 | Page 302

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

Exercise 13.1 | Q 17 | Page 302

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

Exercise 13.1 | Q 18 | Page 302

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

Exercise 13.1 | Q 19 | Page 302

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

Exercise 13.1 | Q 20 | Page 302

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

Exercise 13.1 | Q 21 | Page 302

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

Exercise 13.1 | Q 22 | Page 302

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

Exercise 13.1 | 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):}

Exercise 13.1 | Q 24 | Page 302

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

Exercise 13.1 | Q 25 | Page 302

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

Exercise 13.1 | Q 26 | Page 302

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

Exercise 13.1 | Q 27 | Page 302

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

Exercise 13.1 | 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?

Exercise 13.1 | 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)

Exercise 13.1 | 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?

Exercise 13.1 | 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)

Exercise 13.1 | 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?

Exercise 13.2 [Pages 312 - 313]

### NCERT solutions for Class 11 Mathematics Chapter 13 Limits and Derivatives Exercise 13.2 [Pages 312 - 313]

Exercise 13.2 | Q 1 | Page 312

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

Exercise 13.2 | Q 2 | Page 312

Find the derivative of 99x at x = 100.

Exercise 13.2 | Q 3 | Page 312

Find the derivative of at = 1.

Exercise 13.2 | Q 4.1 | Page 312

Find the derivative of the following functions from first principle.

x3 – 27

Exercise 13.2 | Q 4.2 | Page 312

Find the derivative of the following functions from first principle.

(x – 1) (– 2)

Exercise 13.2 | Q 4.3 | Page 312

Find the derivative of the following functions from first principle.

1/x^2

Exercise 13.2 | Q 4.4 | Page 312

Find the derivative of the following functions from first principle.

(x+1)/(x -1)

Exercise 13.2 | 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)

Exercise 13.2 | 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

Exercise 13.2 | Q 7.1 | Page 313

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

Exercise 13.2 | Q 7.2 | Page 313

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

Exercise 13.2 | Q 7.3 | Page 313

For some constants a and b, find the derivative of

(x-a)/(x -b)

Exercise 13.2 | Q 8 | Page 313

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

Exercise 13.2 | Q 9.1 | Page 313

Find the derivative of 2x - 3/4

Exercise 13.2 | Q 9.2 | Page 313

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

Exercise 13.2 | Q 9.3 | Page 313

Find the derivative of x–3 (5 + 3x)

Exercise 13.2 | Q 9.4 | Page 313

Find the derivative of x5 (3 – 6x–9)

Exercise 13.2 | Q 9.5 | Page 313

Find the derivative of x–4 (3 – 4x–5)

Exercise 13.2 | Q 9.6 | Page 313

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

Exercise 13.2 | Q 10 | Page 313

Find the derivative of cos x from first principle.

Exercise 13.2 | Q 11.1 | Page 313

Find the derivative of the following functions

sin x cos x

Exercise 13.2 | Q 11.2 | Page 313

Find the derivative of the following functions: sec x

Exercise 13.2 | Q 11.3 | Page 313

Find the derivative of the following functions:

5 sec x + 4 cos x

Exercise 13.2 | Q 11.4 | Page 313

Find the derivative of the following functions:

cosec x

Exercise 13.2 | Q 11.5 | Page 313

Find the derivative of the following functions:

3cot x + 5cosec x

Exercise 13.2 | Q 11.6 | Page 313

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

Exercise 13.2 | Q 11.7 | Page 313

Find the derivative of the following functions:

2tan x – 7sec x

Miscellaneous Exercise [Pages 317 - 318]

### NCERT solutions for Class 11 Mathematics Chapter 13 Limits and Derivatives Miscellaneous Exercise [Pages 317 - 318]

Miscellaneous Exercise | Q 1.1 | Page 317

Find the derivative of the following functions from first principle:

−x

Miscellaneous Exercise | Q 1.2 | Page 317

Find the derivative of the following functions from first principle:

(–x)–1

Miscellaneous Exercise | Q 1.3 | Page 317

Find the derivative of the following functions from first principle:

sin (x + 1)

Miscellaneous Exercise | Q 1.4 | Page 317

Find the derivative of the following functions from first principle:

cos (x - pi/8)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | Q 4 | 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) (cx + d)2

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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)

Miscellaneous Exercise | 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

Exercise 13.1Exercise 13.2Miscellaneous Exercise

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

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

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Concepts covered in Class 11 Mathematics 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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