# Question-1

Find the indicated limit:

Solution:
= by direct substitution
= =

# Question-2

Find the indicated limit:

Solution:
= = = 0

# Question-3

Find the indicated limit:

Solution:
= = = = 2x

# Question-4

Find the indicated limit:

Solution:
= = m(1)m â€“ 1 = m

# Question-5

Find the indicated limit:

Solution:
= Ã—

=

=

=

= 2

=

# Question-6

Find the indicated limit:

Solution:

= Ã—

= Ã—

= =

# Question-7

Find the indicated limit:

Solution:
= = (nan â€“ 1 formula)

# Question-8

Find the indicated limit:

Solution:
=

= =

= = =

# Question-9

Find the indicated limit:

Solution:
= Ã— =
=

= =

# Question-10

Find the indicated limit:

Solution:
= = 1 Ã— =

# Question-11

Find the indicated limit:

Solution:
= = 2 cos a = 2 cos

# Question-12

Find the indicated limit:

Solution:
= =

# Question-13

Find the indicated limit:

Solution:
= = e. (1)5 = e

# Question-14

Evaluate the left and right limits of f(x) = at x = 3. Does the limit of f(x) = x â†’ 3 exist? Justify your answer.

Solution:

Let x = 3 + h

Then = = = 27

Let x = 3 â€“ h

Then = = 27

Also, = = 9 + 9 + 9 = 27

# Question-15

Find the positive integer n such that = 108.

Solution:
= 108

n 3n â€“ 1 = 108

Put n = 4, then 4.33 = 4 Ã— 27 = 108

âˆ´ n = 4

# Question-16

Evaluate [Hint: Take ex or esinx as common factor in numerator]

Solution:
=

= =

= ex

= ex

= ex [ 1 â€“ (x - sin x) â€¦â€¦.] = e0 = 1

# Question-17

If f(x) = , f(x) = 1 and f(x) = 1

Solution:
f(x) = = b/-1 = -b = 1
âˆ´ b = -1
f(x) = = a = 1
âˆ´ a = 1

âˆ´ f(x) = = 1

f(2) = 1 and f(-2) = 1

# Question-18

Evaluate and . What can you say about ?

Solution:
Let f(x) = where = x            x â‰¥ 0
= -x           x < 0

Then, f(x) = = = 1

f(x) = = = -1

âˆ´ f(x) â‰  f(x)

âˆ´ does not exist.

# Question-19

Compute Hence evaluate .

Solution:
= = = log a â€“ log b = log
âˆ´ = log

# Question-20

Without using the series expansion of log(1 + x), prove that = 1

Solution:

Let y = log(1 + x) Then as x â†’ 0 , y â†’ 0
= = = = 1

# Question-21

Differentiate the following with respect to x:
(i) x7 + ex

(ii) log7x + 200
(iii) 3 sinx + 4 cos x â€“ ex
(iv) ex + 3 tanx + log x6
(v) sin 5 + log10x + 2 sec x
(vi) x-3/2 + 8e + 7tanx
(vii)
(viii)

Solution:
(i) y = x7 + ex
= 7x6 +ex

(ii) y = log7x + 200
= logex . log10e + 200
= log10e

(iii) y = 3 sin x + 4 cos x â€“ ex
= 3 cos x - 4 sin x â€“ ex

(iv) y = ex + 3 tanx + 6 log x
= ex + 3 sec2x +

(v) y = sin 5 + log10x + 2 sec x
= sin 5 + logex log10e + 2 sec x
= 0 +

(vi) y = x-3/2 + 8e + 7tanx
= -

(vii) y = = x3 + 3x + = x3 + 3x + 3x-1 + x-3
= 3x2 + 3 â€“ 3x-2 - 3x-4

(viii) =

y = 2x2 â€“ 6x â€“ 4 +
= 4x â€“ 6 -

# Question-22

Differentiate the following function using quotient rule.

Solution:
Let y =

= = -

# Question-23

Differentiate the following function using quotient rule.

Solution:
Let y =

= =

# Question-24

Differentiate the following function using quotient rule.

Solution:
Let y =

=

=

=
=

# Question-25

Differentiate the following function using quotient rule.

Solution:
Let y = =
= = = =
e-x

# Question-26

Differentiate the following function with respect to x.
log(sinx)

Solution:
y = log(sinx)

Let u = sinx

= cosx

y = log u

= =

= Ã— = .cos x = cos x = cot x

# Question-27

Differentiate the following function with respect to x.
esinx

Solution:
y = esinx

Put u = sinx

= cosx

y = eu

= eu = esinx

âˆ´ = . = esinx. cosx

# Question-28

Differentiate the following function with respect to x.

Solution:
y =

Put u = 1 + cotx

= -cosec2 x

y = u1/2

= =

âˆ´ = =

# Question-29

Differentiate the following function with respect to x.
tan(logx)

Solution:
y = tan(logx)

Put u = logx

=

y = tan u

= sec2 u

âˆ´ = . = sec2 u. =

# Question-30

Differentiate the following function with respect to x.

Solution:
y =

=

=

=

# Question-31

Differentiate the following function with respect to x.
log sec

Solution:
y = log sec
Put u =
=

y = log sec u

y = log v

Put v = sec u

= sec u tan u

=

âˆ´ = . .

= . sec tan .
=
=

# Question-32

Differentiate the following function with respect to x.
log sin(ex + 4x + 5)

Solution:
y = log sin(ex + 4x + 5)

=

= (ex + 4)

= (ex + 4)cot (ex + 4x + 5)

# Question-33

Differentiate the following function with respect to x.
sin(x3/2)

Solution:
y = sin(x3/2)

Put u = x3/2

y = sin u

=

= cos u
= . = cos u. x1/2 = cos x3/2

# Question-34

Differentiate the following function with respect to x.
cos()

Solution:
y = cos u

Put u =

= -sin u

=

âˆ´ = . = - sin u . =

# Question-35

Differentiate the following function with respect to x.
esin(log x)

Solution:
y = esin(log x)

Put u = logx

=

y = esinu

Put v = sin u

= cos u = cos(logx)

Put y = ev

= ev
Hence = . = ev . cos(logx). = esin(logx) cos(logs) .

# Question-36

Find the indicated limit:

Solution:
= by direct substitution

= = 4

# Question-37

Find the indicated limit:

Solution:
= = = 0

# Question-38

Find the indicated limit:

Solution:
= = = = 2x

# Question-39

Find the indicated limit:

Solution:
= = m(1)m â€“ 1 = m

# Question-40

Find the indicated limit:

Solution:
= Ã— =

=

=

= 2

=

# Question-41

Find the indicated limit:

Solution:

= Ã—

= Ã—

= =

# Question-42

Find the indicated limit:

Solution:
= =    (nan â€“ 1 formula)

# Question-43

Find the indicated limit:

Solution:
=

= =

= = =

# Question-44

Find the indicated limit:

Solution:
= Ã— =
=

= =

# Question-45

Find the indicated limit:

Solution:
= = 1 Ã— =

# Question-46

Find the indicated limit:

Solution:
= = 2 cos a = 2 cos a

# Question-47

Find the indicated limit:

Solution:
= =

# Question-48

Find the indicated limit:

Solution:
= = e. (1)5 = e

# Question-49

Evaluate the left and right limits of f(x) = at x = 3. Does the limit of f(x) = x â†’ 3 exist? Justify your answer.

Solution:

Let x = 3 + h

Then = = = 27

Let x = 3 â€“ h

Then = = 27

Also, = = 9 + 9 + 9 = 27

# Question-50

Find the positive integer n such that = 108.

Solution:
= 108

n 3n â€“ 1 = 108

Put n = 4, then 4.33 = 4 Ã— 27 = 108

âˆ´ n = 4

# Question-51

Evaluate [Hint: Take ex or esinx as common factor in numerator]

Solution:
=

= =

= ex

= ex

= ex [ 1 â€“ (x - sin x) â€¦â€¦.] = e0 = 1

# Question-52

If f(x) = , f(x) = 1 and f(x) = 1

Solution:
f(x) = = b/-1 = -b = 1
âˆ´ b = -1
f(x) = = a = 1
âˆ´ a = 1

âˆ´ f(x) = = 1

f(2) = 1 and f(-2) = 1

# Question-53

Evaluate and . What can you say about ?

Solution:
Let f(x) = where = x            x â‰¥ 0
= -x           x < 0

Then,

f(x) = = = 1

f(x) = = = -1
âˆ´ f(x) â‰  f(x)

âˆ´ does not exist.

# Question-54

Compute Hence evaluate .

Solution:
= = = log a â€“ log b = log

âˆ´ = log

# Question-55

Without using the series expansion of log(1 + x), prove that = 1

Solution:

Let y = log(1 + x) Then as x â†’ 0 , y â†’ 0

=  = = 1

# Question-56

Differentiate the following with respect to x:
(i) x7 + ex

(ii) log7x + 200
(iii) 3 sinx + 4 cos x â€“ ex
(iv) ex + 3 tanx + log x6
(v) sin 5 + log10x + 2 sec x
(vi) x-3/2 + 8e + 7tanx
(vii)
(viii)

Solution:
(i) y = x7 + ex

= 7x6 +ex

(ii) y = log7x + 200 = logex . log10e + 200

= log10e

(iii) y = 3 sin x + 4 cos x â€“ ex

= 3 cos x - 4 sin x â€“ ex

(iv) y = ex + 3 tanx + 6 log x

= ex + 3 sec2x +

(v) y = sin 5 + log10x + 2 sec x

= sin 5 + logex log10e + 2 sec x

= 0 +

(vi) y = x-3/2 + 8e + 7tanx

= -

(vii) y = = x3 + 3x + = x3 + 3x + 3x-1 + x-3

= 3x2 + 3 â€“ 3x-2 - 3x-4

(viii) =

y = 2x2 â€“ 6x â€“ 4 +

= 4x â€“ 6 -

# Question-57

Differentiate the following functions with respect to x.
(i) ex cos x
(ii)
(iii) 6 sin x log10 x + e
(iv) (x4 â€“ 6x3 + 7x2 + 4x + 2) (x3 â€“ 1)
(v) (a â€“ b six (1 â€“ 2 cos x)
(vi) cosec x . cotx
(vii) sin2 x
(viii) cos2 x
(ix) (3x2 + 1)2
(x) (4x2 â€“ 1) (2x + 3)
(xi) (3 sec x â€“ 4 cosec x) (2 sin x + 5 cos x)
(xii) x2 ex sin x
(xiii) ex cos x

Solution:
(i) y = ex cos x

= - ex sin x + cos x ex

(ii) y = x1/n log() = x1/n log x
= =
(iii) y = 6 sin x log10 x + e = 6 sin x logex . log10 e + e

= 6 log10 e

(iv) y = (x4 â€“ 6x3 + 7x2 + 4x + 2)(x3 â€“ 1)

= (x4 â€“ 6x3 + 7x2 + 4x + 2) (3x2) + (x3 â€“ 1) (4x3 â€“ 18x2 + 14x + 4)

(v) y = (a â€“ b six (1 â€“ 2 cos x)

= (a â€“ b sinx) (2 sin x) + (1 â€“ 2 cos x) (-b cosx)

= 2a sinx â€“ 2b sin2 x â€“ b cos x + 2b cos2x

(vi) y = cosec x. cot x

= -cosec x cosec2x(-cosec x cotx) = -cosec3 x â€“ cot2x cosec x

(vii) y = sin2x = sin x. sin x

= sinx cosx + cosx sinx = 2 sin x cos x = sin 2x

(viii) y = cos2x = cos x cos x

= - cos x sin x - cos x sin x = -2 sin x cos x = -sin 2x

(ix) y = (3x2 +1) (3x2 +1)

= (3x2 +1) 6x + (3x2 +1) 6x = 12x (3x2 +1)

(x) y = (4x2 â€“ 1) (2x + 3)

= (4x2 â€“ 1) (2x + 3)
= (4x2 â€“ 1) (2) + (2x + 3) 8x = 8x2 â€“ 2 + 16x2 + 24x

= 24x2 + 24x - 2 = 2(12x2 + 12 x â€“ 1)

(xi) y = (3 sec x â€“ 4 cosec x) (2 sin x + 5 cos x)

= (3 sec x â€“ 4 cosec x) (2 cos x â€“ 5 sin x) + (2 sin x tan x + 4 cosec x cot x)

= 6 sec x cos x - 8

= 6 â€“ 8 cot x - 15 tan x + 20 + 6
= 26 + 6 = 26 + 6 tan + 20 cot2x

(xii) y = x2 ex sinx

= x2 ex cos x + ex sinx(2x) + x2 sin x ex

(xiii) y = ex cos x

= ex . + logx ex + ex log x

# Question-58

Differentiate the following function using quotient rule.

Solution:
Let y =
= = = =

# Question-59

Differentiate the following function using quotient rule.

Solution:
Let y =

=

=

=

=

# Question-60

Differentiate the following function using quotient rule.

Solution:
Let y =
= =

# Question-61

Differentiate the following function using quotient rule.

Solution:
Let y =

=

=

# Question-62

Differentiate the following function using quotient rule.

Solution:
Let y =

= =

# Question-63

Differentiate the following function using quotient rule.

Solution:
Let y =
=

# Question-64

Differentiate the following function using quotient rule.

Solution:
Let y =

= = -

# Question-65

Differentiate the following function using quotient rule.

Solution:
Let y =

= =

# Question-66

Differentiate the following function using quotient rule.

Solution:
Let y =

=

=

=
=

# Question-67

Differentiate the following function using quotient rule.

Solution:
Let y = =

= = = =
e-x

# Question-68

Differentiate the following function with respect to x.

Solution:
y = log(sinx)

Let u = sinx

= cosx

y = log u

= =

= Ã— = .cos x = cos x = cot x

# Question-69

Differentiate the following function with respect to x.
esinx

Solution:
y = esinx

Put u = sinx

= cosx

y = eu

= eu = esinx

âˆ´ = . = esinx. cosx

# Question-70

Differentiate the following function with respect to x.

Solution:
y =

Put u = 1 + cotx

= -cosec2 x

y = u1/2

=

âˆ´ = =

# Question-71

Differentiate the following function with respect to x.
tan(logx)

Solution:
y = tan(logx)

Put u = logx

=

y = tan u

= sec2 u

âˆ´ = . = sec2 u. =

# Question-72

Differentiate the following function with respect to x.

Solution:
y =

=

=

=

# Question-73

Differentiate the following function with respect to x.
log sec

Solution:
y = log sec
Put u =

=

y = log sec u

y = log v

Put v = sec u

= sec u tan u

=

âˆ´ = . .

= . sec tan .
=
=

# Question-74

Differentiate the following function with respect to x.
log sin(ex + 4x + 5)

Solution:
y = log sin(ex + 4x + 5)

=

= (ex + 4)

= (ex + 4)cot (ex + 4x + 5)

# Question-75

Differentiate the following function with respect to x.
sin(x3/2)

Solution:
y = sin(x3/2)

Put u = x3/2

y = sin u

=

= cos u

= . = cos u. x1/2 = cos x3/2

# Question-76

Differentiate the following function with respect to x.
cos()

Solution:
y = cos u

Put u =

= -sin u

=

âˆ´ = . = - sin u . =

# Question-77

Differentiate the following function with respect to x.
esin(log x)

Solution:
y = esin(log x)

Put u = logx

=

y = esinu

Put v = sin u

= cos u = cos(logx)

Put y = ev

= ev
Hence = . = ev . cos(logx). = esin(logx) cos(logs) .