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

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Chapter 13 - Limits and Derivatives

Pages 301 - 303

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

Q 1 | Page 301 |

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

Q 2 | Page 301 |

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

Q 3 | Page 301 |

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

Q 4 | Page 301 |

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

Q 5 | Page 301 |

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

Q 6 | Page 301 |

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

Q 7 | Page 301 |

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

Q 8 | Page 301 |

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

Q 9 | Page 301 |

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

Q 10 | Page 301 |

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

Q 11 | Page 301 |

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

Q 12 | Page 301 |

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

Q 13 | Page 301 |

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

Q 14 | Page 301 |

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

Q 15 | Page 302 |

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

Q 16 | Page 302 |

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

Q 17 | Page 302 |

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

Q 18 | Page 302 |

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

Q 19 | Page 302 |

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

Q 20 | Page 302 |

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

Q 21 | Page 302 |

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

Q 22 | 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 23 | Page 302 |

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

Q 24 | Page 302 |

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

Q 25 | Page 302 |

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

Q 26 | Page 302 |

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

Q 27 | 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 28 | Page 302 |

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 29 | 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 30 | 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 31 | 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?

Q 32 | Page 303 |

Pages 312 - 313

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

Q 1 | Page 312 |

Find the derivative of 99x at x = 100.

Q 2 | Page 312 |

Find the derivative of at = 1.

Q 3 | Page 312 |

Find the derivative of the following functions from first principle.

x3 – 27

Q 4.1 | Page 312 |

Find the derivative of the following functions from first principle.

(x – 1) (– 2)

Q 4.2 | Page 312 |

Find the derivative of the following functions from first principle.

`1/x^2`

Q 4.3 | Page 312 |

Find the derivative of the following functions from first principle.

`(x+1)/(x -1)`

Q 4.4 | 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 5 | Page 312 |

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 6 | Page 313 |

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

 

Q 7.1 | Page 313 |

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

Q 7.2 | Page 313 |

For some constants a and b, find the derivative of

`(x-a)/(x -b)`

Q 7.3 | Page 313 |

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

Q 8 | Page 313 |

Find the derivative of `2x - 3/4`

Q 9.1 | Page 313 |

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

Q 9.2 | Page 313 |

Find the derivative of x–3 (5 + 3x)

Q 9.3 | Page 313 |

Find the derivative of x5 (3 – 6x–9)

Q 9.4 | Page 313 |

Find the derivative of x–4 (3 – 4x–5)

Q 9.5 | Page 313 |

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

Q 9.6 | Page 313 |

Find the derivative of cos x from first principle.

Q 10 | Page 313 |

Find the derivative of the following functions

sin x cos x 

Q 11.1 | Page 313 |

Find the derivative of the following functions: sec x

Q 11.2 | Page 313 |

Find the derivative of the following functions:

5 sec x + 4 cos x

Q 11.3 | Page 313 |

Find the derivative of the following functions:

cosec x

Q 11.4 | Page 313 |

Find the derivative of the following functions:

3cot x + 5cosec x

Q 11.5 | Page 313 |

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

Q 11.6 | Page 313 |

Find the derivative of the following functions:

2tan x – 7sec x

Q 11.7 | Page 313 |

Pages 317 - 318

Find the derivative of the following functions from first principle:

−x

Q 1.1 | Page 317 |

Find the derivative of the following functions from first principle:

(–x)–1

Q 1.2 | Page 317 |

Find the derivative of the following functions from first principle:

sin (x + 1)

Q 1.3 | Page 317 |

Find the derivative of the following functions from first principle: 

`cos (x - pi/8)`

Q 1.4 | 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 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): `(px+ q) (r/s + s)`

Q 3 | 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 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)`

Q 5 | 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 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/(ax^2 + bx + c)`

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

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

Q 9 | 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 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):  `4sqrtx - 2`

Q 11 | 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 12 | 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 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): sin (x + a)

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): cosec x cot x

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): `cos x/(1 +  sin 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) : `(sin x + cos x)/(sin x - cos 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) :  `(sec x - 1)/(sec x + 1)`

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): sinn x

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

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

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): x4 (5 sin x – 3 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): (x2 + 1) 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): (ax2 + sin x) (p + q 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): `(x + cos x)(x - tan 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): `(4x + 5sin x)/(3x + 7cos 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):  `(x^2 cos (pi/4))/sin 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/(1 + tan 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 + sec x) (x – 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/(sin^n x)`

Q 30 | Page 318 |

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