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

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Solutions for Chapter 13: Limits and Derivatives

Below listed, you can find solutions for Chapter 13 of CBSE, Karnataka Board PUC NCERT for Mathematics Class 11.


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

NCERT solutions for Mathematics Class 11 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 Mathematics Class 11 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 Mathematics Class 11 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)`

Solutions for Chapter 13: Limits and Derivatives

Exercise 13.1Exercise 13.2Miscellaneous Exercise

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

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Mathematics Class 11 CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Mathematics Class 11 CBSE, Karnataka Board PUC 13 (Limits and Derivatives) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

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

Concepts covered in Mathematics 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, 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, Limits of Logarithmic Functions, 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, 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, Limits of Logarithmic Functions.

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

Get the free view of Chapter 13, Limits and Derivatives Mathematics Class 11 additional questions for Mathematics Mathematics Class 11 CBSE, Karnataka Board PUC, and you can use Shaalaa.com to keep it handy for your exam preparation.

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