# 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**

^{m â€“ 1}=# Question-5

**Find the indicated limit:**

**Solution:**

**= Ã—**

=

=

=

=

** = **

** = **2

=

# Question-6

**Find the indicated limit:**

**Solution:**

= Ã—

= Ã—

= Ã—

= Ã—

**= = **

# Question-7

**Find the indicated limit:**

**Solution:**

**= =**(na

^{n 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 3^{n â€“ 1} = 108

Put n = 4, then 4.3^{3} = 4 Ã— 27 = 108

âˆ´ n = 4

# Question-16

**Evaluate [Hint: Take e**

^{x}or e^{sinx}as common factor in numerator]**Solution:**

=

= =

= e

^{x}

= e^{x}

= e^{x} [ 1 â€“ (x - sin x) â€¦â€¦.] = e^{0} = 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) x

(i) x

^{7 + }e^{x}**(ii) log**

(iii) 3 sinx + 4 cos x â€“ e

(iv) e

(v) sin 5 + log

(vi) x

(vii)

(viii)

_{7}x + 200(iii) 3 sinx + 4 cos x â€“ e

^{x}(iv) e

^{x}+ 3 tanx + log x^{6}(v) sin 5 + log

_{10}x + 2 sec x(vi) x

^{-3/2}+ 8e + 7tanx(vii)

(viii)

**Solution:**

(i) y = x

^{7}+ e

^{x}

**=**7x

^{6}+e

^{x}

(ii) y = log

_{7}x + 200

= log

_{e}x . log

_{10}e + 200

**=**log

_{10}e

(iii) y = 3 sin x + 4 cos x â€“ e^{x}

** = **3 cos x - 4 sin x â€“ e^{x}

(iv) y = e^{x} + 3 tanx + 6 log x

** = **e^{x} + 3 sec^{2}x +

(v) y = sin 5 + log_{10}x + 2 sec x

= sin 5 + log_{e}x log_{10}e + 2 sec x

= 0 +

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

= -

(vii)** **y = = x^{3} + 3x + = x^{3} + 3x + 3x^{-1} + x^{-3}

= 3x^{2} + 3 â€“ 3x^{-2} - 3x^{-4}

(viii) ** =
**y = 2x

^{2}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)

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.**

e

e

^{sinx}**Solution:**

y = e

^{sinx}

Put

**u = sinx**

= cosx

y = e

^{u}

= e

^{u}= e

^{sinx}

âˆ´ = . = e^{sinx}. cosx

# Question-28

**Differentiate the following function with respect to x.**

**Solution:**

y =

Put** **u = 1 + cotx

= -cosec^{2} x

y = u^{1/2 }

= =

âˆ´ = =

# Question-29

**Differentiate the following function with respect to x.**

tan(logx)

tan(logx)

**Solution:**

y = tan(logx)

Put u = logx

=

y = tan u

= sec^{2} u

âˆ´ = . = sec^{2} 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

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(e

log sin(e

^{x}+ 4x + 5)**Solution:**

y = log sin(e

^{x}+ 4x + 5)

=

= (e^{x} + 4)

= (e^{x} + 4)cot (e^{x} + 4x + 5)

# Question-33

**Differentiate the following function with respect to x.**

sin(x

sin(x

^{3/2})**Solution:**

y = sin(x

^{3/2})

Put u = x

^{3/2}

y = sin u

=

= cos u

= . = cos u. x

^{1/2}= cos x

^{3/2}

# Question-34

**Differentiate the following function with respect to x.**

cos()

cos()

**Solution:**

y = cos u

Put u =

= -sin u

=

âˆ´ = . = - sin u . =

# Question-35

**Differentiate the following function with respect to x.**

e

e

^{sin(log x)}**Solution:**

y = e

^{sin(log x) }

Put u = logx

=

y = e

^{sinu }

Put v = sin u

= cos u = cos(logx)

Put y = e

^{v}

= e

^{v }

Hence = . = e

^{v}. cos(logx). = e

^{sin(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**

^{m â€“ 1}=# 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:**

**= =**(na

^{n 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 3^{n â€“ 1} = 108

Put n = 4, then 4.3^{3} = 4 Ã— 27 = 108

âˆ´ n = 4

# Question-51

**Evaluate [Hint: Take e**

^{x}or e^{sinx}as common factor in numerator]**Solution:**

=

= =

= e

^{x}

= e^{x}

= e^{x} [ 1 â€“ (x - sin x) â€¦â€¦.] = e^{0} = 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) x

(i) x

^{7 + }e^{x}**(ii) log**

(iii) 3 sinx + 4 cos x â€“ e

(iv) e

(v) sin 5 + log

(vi) x

(vii)

(viii)

_{7}x + 200(iii) 3 sinx + 4 cos x â€“ e

^{x}(iv) e

^{x}+ 3 tanx + log x^{6}(v) sin 5 + log

_{10}x + 2 sec x(vi) x

^{-3/2}+ 8e + 7tanx(vii)

(viii)

**Solution:**

(i) y = x

^{7}+ e

^{x }

**=**7x

^{6}+e

^{x}

(ii) y = log

_{7}x + 200 = log

_{e}x . log

_{10}e + 200

** = **log_{10}e

(iii) y = 3 sin x + 4 cos x â€“ e^{x}

** = **3 cos x - 4 sin x â€“ e^{x}

(iv) y = e^{x} + 3 tanx + 6 log x

** = **e^{x} + 3 sec^{2}x +

(v) y = sin 5 + log_{10}x + 2 sec x

= sin 5 + log_{e}x log_{10}e + 2 sec x

= 0 +

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

= -

(vii)** **y = = x^{3} + 3x + = x^{3} + 3x + 3x^{-1} + x^{-3}

= 3x^{2} + 3 â€“ 3x^{-2} - 3x^{-4}

(viii) ** =
**y = 2x

^{2}â€“ 6x â€“ 4 +

= 4x â€“ 6 -

# Question-57

**Differentiate the following functions with respect to x.**

(i) e

(ii)

(iii) 6 sin x log

(iv) (x

(v) (a â€“ b six (1 â€“ 2 cos x)

(vi) cosec x . cotx

(vii) sin

(viii) cos

(ix) (3x

(x) (4x

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

(xii) x

(xiii) e

(i) e

^{x}cos x(ii)

(iii) 6 sin x log

_{10}x + e(iv) (x

^{4}â€“ 6x^{3}+ 7x^{2}+ 4x + 2) (x^{3}â€“ 1)(v) (a â€“ b six (1 â€“ 2 cos x)

(vi) cosec x . cotx

(vii) sin

^{2 }x(viii) cos

^{2}x(ix) (3x

^{2}+ 1)^{2}(x) (4x

^{2}â€“ 1) (2x + 3)(xi) (3 sec x â€“ 4 cosec x) (2 sin x + 5 cos x)

(xii) x

^{2}e^{x}sin x(xiii) e

^{x}cos x**Solution:**

(i) y = e

^{x}cos x

= - e

^{x}sin x + cos x e

^{x}

(ii) y = x

^{1/n}log() = x

^{1/n}log x

= =

(iii) y = 6 sin x log

_{10}x + e = 6 sin x log

_{e}x . log

_{10}e + e

= 6 log

_{10}

^{ }e

(iv) y = (x^{4} â€“ 6x^{3} + 7x^{2} + 4x + 2)(x^{3} â€“ 1)

= (x^{4} â€“ 6x^{3} + 7x^{2} + 4x + 2) (3x^{2}) + (x^{3} â€“ 1) (4x^{3} â€“ 18x^{2} + 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 sin^{2} x â€“ b cos x + 2b cos^{2}x

(vi) y = cosec x. cot x

= -cosec x cosec^{2}x(-cosec x cotx) = -cosec^{3 }x â€“ cot^{2}x cosec x

(vii) y = sin^{2}x = sin x. sin x

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

(viii) y = cos^{2}x = cos x cos x

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

(ix) y = (3x^{2 }+1) (3x^{2 }+1)

= (3x^{2 }+1) 6x + (3x^{2 }+1) 6x = 12x (3x^{2 }+1)

(x) y = (4x^{2} â€“ 1) (2x + 3)

= (4x^{2} â€“ 1) (2x + 3)

= (4x^{2} â€“ 1) (2) + (2x + 3) 8x = 8x^{2} â€“ 2 + 16x^{2} + 24x

= 24x^{2} + 24x - 2 = 2(12x^{2} + 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 cot^{2}x

(xii) y = x^{2} e^{x} sinx

= x^{2} e^{x} cos x + e^{x} sinx(2x) + x^{2} sin x e^{x}

(xiii) y = e^{x} cos x

= e^{x} . + log_{x} e^{x} + e^{x} 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.**

e

e

^{sinx}**Solution:**

y = e

^{sinx}

Put

**u = sinx**

= cosx

y = e

^{u}

= e

^{u}= e

^{sinx}

âˆ´ = . = e^{sinx}. cosx

# Question-70

**Differentiate the following function with respect to x.**

**Solution:**

y =

Put** **u = 1 + cotx

= -cosec^{2} x

y = u^{1/2 }

= =

âˆ´ = =

# Question-71

**Differentiate the following function with respect to x.**

tan(logx)

tan(logx)

**Solution:**

y = tan(logx)

Put u = logx

=

y = tan u

= sec^{2} u

âˆ´ = . = sec^{2} 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

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(e

log sin(e

^{x}+ 4x + 5)**Solution:**

y = log sin(e

^{x}+ 4x + 5)

=

= (e^{x} + 4)

= (e^{x} + 4)cot (e^{x} + 4x + 5)

# Question-75

**Differentiate the following function with respect to x.**

sin(x

sin(x

^{3/2})**Solution:**

y = sin(x

^{3/2})

Put u = x

^{3/2}

y = sin u

=

= cos u

= . = cos u. x

^{1/2}= cos x

^{3/2}

# Question-76

**Differentiate the following function with respect to x.**

cos()

cos()

**Solution:**

y = cos u

Put u =

= -sin u

=

âˆ´ = . = - sin u . =

# Question-77

**Differentiate the following function with respect to x.**

e

e

^{sin(log x)}**Solution:**

y = e

^{sin(log x) }

Put u = logx

=

y = e

^{sinu }

Put v = sin u

= cos u = cos(logx)

Put y = e

^{v}

= e

^{v }

Hence = . = e

^{v}. cos(logx). = e

^{sin(logx)}cos

^{(logs)}.