# CLASS 11 MATHS CHAPTER-13 LIMITS AND DERIVATIVES

### Evaluate the following limits in Exercises 1 to 22.

Question 1.

Solution :

Given that,

Substituting x = 3, we get

= 3 + 3 = 6

Question 2.

Solution :

Given that,

Substituting x = π, we get
=(π – 22 / 7)

Question 3.

Solution :

Given that,

Substituting r = 1, we get
π(1)2 = π

Question 4.

Solution :
given limit,

Substituting x = 4, we get

= [4(4) + 3] / (4 – 2) =

= (16 + 3) / 2

= 19 / 2

Question5.

Solution :

Given that,

Substituting x = -1, we get

= [(-1)10 + (-1)5 + 1] / (-1 – 1)

= (1 – 1 + 1) / – 2

= – 1 / 2

Question6.

Solution :
Given limit,

Putting x as 0

= [(0 + 1)5 – 1] / 0

=0

Since, this limit is undefined

Substitute x + 1 = y, then x = y – 1

=5

Question 7.

Solution :

Question 8.

Solution :

Question 9.

Solution :
Given that,

= [a (0) + b] / c (0) + 1

= b / 1

= b

Question 10.

Solution :

Question 11.

Solution :
Given that,

Substituting x = 1

= [a (1)2 + b (1) + c] / [c (1)2 + b (1) + a]

= (a + b + c) / (a + b + c)

= 1

Question 12.

Solution :

Question 13.

Solution :
Given that

Question 14.

Solution :

Question 15.

Solution :

Question 16.

Solution :

Question 17.

Solution :

Question18.

Solution :

Question19.

Solution :

Question20.

Solution :

Question21.

Solution :

Question22.

Solution :

= 1 x 2

= 2

Question23.Find and where

Solution :

Question 24. Find where

Solution :
Given:

.

Question 25. Evaluate where

Solution :
Given:

.

Question26. Find Where

Solution :
Given:

Question 27. Find where

Solution :
Given:

Question 28. Suppose and if what are possible values of a and b?

Solution :
Given: and

Question 29. Let a1, a2,………abe fixed real numbers and define a function f (x) = (x – a1) (x – a2) ……. (x – an).What is  For some a ≠ a1, a2, ……. an, compute

Solution :

Question30. If for what values of (s) does exists?

Solution :

Question 31. If the function f(x) satisfies valuate

Solution :

Question32. If for what integer m and n does both and exist?

Solution :
Given function is

### Evaluate the following limits in Exercises 1 to 22.

Question1. Find the derivative of x2 – 2 at x = 10.

Solution :
Here f(x) = x2 – 2

Question2.Find the derivative of 99x at x = 100.

Solution :

Let f (x) = x

Then,

Question3. Find the derivative of x at x=1.

Solution :

Let f (x) = 99x,

From first principle

Question4. Find the derivatives of the following functions from first principle:

(i) x3−27

(ii) (x−1)(x−2)

(iii)1/x2

( iv )x+1/x−1

Solution :
(i)Let f (x) = x3 – 27

From first principle

(ii) Let f (x) = (x – 1) (x – 2)

From first principle

(iii)Let f (x) = 1 / x2

From first principle, we get

(iv)Let f (x) = x + 1 / x – 1

From first principle, we get

Question5. For the functionprove that f’ (1) =100 f’ (0).

Solution :

Question6. Find the derivative of xn + axn−1 + a2xn−2 + …+an−1x + an for some fixed real number a.

Solution :

Question7. For some constants a and b find the derivative of:

(i) (x − a) (x − b)

(ii) (ax2 + b)2

(iii) x – a / x – b

Solution :
(i) Given:(x − a) (x − b)

(ii)(ax2 + b)2

(iii) x – a / x – b

Question8. Find the derivative offor some constant a.

Solution :

Question9. Find the derivative of

(i) 2x−3/4

(ii) (5x3+3x−1)(x−1)

(iii) x−3(5+3x)

(iv) x(3−6x−9)

(iv) x−4(3−4x−5)

(vi) 2/x+1 − x/3x−1

Solution :
(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question10. Find the derivative of cos x from first principle.

Solution :
,

Question11. Find the derivative of the following functions:

(i) sin x cos x

(ii) sec x

(iii) 5sec x + 4cos x

(iv) cosec x

(v) 3cot + 5cosec x

(vi) 5sin – 6cos x + 7

(vii) 2tan x – 7sec x

Solution :
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

### Miscellaneous Exercise

Question1. Find the derivative of the following functions from first principle:

(i) −x

( ii) (−x)−1

( iii) sin(x+1)

(iii)cos(x−π/8)

Solution :
(i)Given:f(x) = -x  then

(ii)Given:

then

(iii)Given:f(x) = sin (x + 1)

then

(iv)Given:

then

Question2. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + a)

Solution :

f(x) = (x + a)

Accordingly,

By first principle,

Question3. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (px+q)(r/x+s)

Solution :
Given:f(x) =

Question4. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):(ax+b)(cx+d)2

Solution :
Given:f(x) = (ax+b)(cx+d)2

Question5. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question6. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question7. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question8. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question9. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question10. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question11. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question12. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax+b)n

Solution :

Question13. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):(ax+b)(cx+d)m

Solution :

since,

Question14. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):sin (x + a)

Solution :

Question15. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x

Solution :

Question16. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question17. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question18. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question19. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sinn

Solution :

Question20. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question21. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question22. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x4(5 sin x – 3 cos x)

Solution :

Question23. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):  (x2+1)cosx

Solution :

Question24. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax+ sinx)(p + qcosx)

Solution :

Question25. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x+cosx) (x−tanx)

Solution :

Question26. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question27. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question28. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question29. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x – tan x)

Solution :
let f(x) = (x + sec x) (x – tan x)

By product rule,

Question30. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

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