Exercise 1.3 (Solutions)



Merging man and maths

Calculus and Analytic Geometry, MATHEMATICS 12

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Important Limits

I.

xn an

lim

xa

xa

nan1,

where

n

is

integer

and

a 0.

II.

lim

x0

xa x

a 1 . 2a

III.

lim

n0

1

1 n

n

e

.

1

IV. lim1 xx e . x

V.

ax 1

lim

x0

x

ln a ,

where

a

0.

VI.

lim

x0

ex

x

1

ln

e

1.

VII.

If

is measured in radian, then

lim

0

sin

1.

Question # 1

(i) lim(2x 4) lim(2x) lim(4) 2lim(x) 4 2(3) 4 10 .

x3

x3

x3

x3

(ii) lim 3x2 2x 4 3(1)2 2(1) 4 3 2 4 5. x1

(iii) lim x2 x 4 (3)2 (3) 4 9 3 4 16 4 . x3

(iv) lim x x2 4 2 22 4 = 0.

x2

(v) lim x3 1 x2 5 lim x3 1 lim x2 5

x2

x2

x2

(2)3 1 (2)2 5

81 45 9 9 0.

(vi) lim 2x3 5x 2(2)3 5(2) 16 10 26 13 .

x2 3x 2

3(2) 2

6 2 8 4

Question # 2

(i) lim x3 x lim x(x2 1) lim x(x 1)(x 1)

x1 x 1

x1 x 1

x1

x 1

lim x(x 1) (1)(11) 2 x1

3x3 4x

(ii)

lim x0

x2 x

x(3x2 4) = lim

x0 x(x 1)

lim 3x2 4 x0 x 1

3(0) 4 4 . 0 1

FSc-II / Ex- 1.3 - 2

(iii)

lim

x2

x3 x2

x

8

6

lim

x2

x2

x3 (2)3 3x 2x

6

lim (x 2)(x2 2x 4) x2 x(x 3) 2(x 3)

lim (x 2)(x2 2x 4) lim (x2 2x 4)

x2 (x 3)(x 2)

x2 (x 3)

(2)2 2(2) 4) 12

(2 3)

5

(iv)

x3 3x2 3x 1

lim

x1

x3 x

x 13

x 13

lim

x1

x(x2

1)

lim x1 x(x 1)(x 1)

x 12

1 12

lim

lim

0

x1 x(x 1) x1 (1) (1 1)

(v)

lim x1

x3 x2 x2 1

x2 x 1

lim x1 (x 1)(x 1)

lim x2 x1 (x 1)

12 1

(1 1) 2

(vi)

lim

x4

2x x3

2 32 4x2

lim

x4

2(x2 16) x2 (x 4)

2(x 4) x 4

lim x4

x2 (x 4)

lim

x4

2(

x x2

4)

2(4 4) 42

16 1. 16

(vii)

x 2 lim x2 x 2

lim x2

x 2

x2

x 2

x 2

2

2

x 2

lim

x2 x 2 x 2

lim

x2

x2 x 2 x 2

(viii) lim h0

xh h

lim 1

1

1

x2 x 2

2 2 2 2

x lim x h x x h x

h0

h

xh x

2

2

x h x

lim

lim

xhx

h0 h x h x

h0 h x h x

FSc-II / Ex- 1.3 - 3

lim

h

h0 h x h x

lim

1

h0 x h x

1

1

x0 x 2 x

(ix)

lim

xa

xn xm

an am

x a xn1 xn2a xn3a2 .... an1 lim

xa x a xm1 xm2a xm3a2 .... am1

xn1 xn2a xn3a2 .... an1 lim

xa xm1 xm2a xm3a2 .... am1

an1 an2a an3a2 am1 am2a am3a2

.... ....

an1 am1

an1 an1 an1 .... an1 am1 am1 am1 .... am1

(n terms) (m terms)

n an1 m am1

n an1m1 n anm

m

m

Law of Sine If is measured in radian, then lim sin 1 0 See proof on book at page 25

Question # 3

(i) lim sin 7x x0 x

Put t 7x t x 7

When x 0 then t 0 , so

lim sin 7x x0 x

lim

t 0

sin t

t

7

7lim sin t 7(1) 7 t0 t

(ii) lim sin x x0 x

Since 180 rad

1 rad 180

So

lim sin x

lim

sin

x 180

x0 x

x0

x

By law of sine. x x rad

180

FSc-II / Ex- 1.3 - 4

Now put x t i.e. x 180t

180

When x 0 then t 0 , so

lim

sin

x 180

lim

sin t

x0

x

x0 180t

lim sin t (1)

180 x0 t

180

180

(iii) lim1 cos lim1 cos 1 cos

0 sin

0 sin 1 cos

lim 1 cos2 lim sin2

0 sin 1 cos 0 sin 1 cos

by law of sine

lim sin sin(0) 0 0

0 1 cos 1 cos(0) 1 1

(iv) lim sin x x x Put t x

x t

When x then t 0 , so

lim sin x lim sin( t)

x x

t 0

t

lim sin t t0 t

1

sin

t

sin

2

2

t

sin t

By law of sine.

(v) lim sin ax limsin ax 1

x0 sin bx

x0

sin bx

lim

x0

sin

ax

ax ax

sin

1 bx

bx bx

lim

x0

sin ax ax

ax

sin

1 bx

bx

bx

a b

lim sin ax x0 ax

lim

1 sin bx

a (1) 1 b (1)

a b

x0 bx

by law of sine

(vi)

lim x x0 tan x

lim

x0

x sin

x

lim x cos x x0 sin x

cos x

lim

x0

1 sin

x

cos

x

x

1 lim sin

x

limcos x0

x

x0 x

11 1 1

(vii)

lim

x0

1

cos x2

2

x

2sin2 x

lim

x0

x2

sin2 x 1 cos 2x 2

2sin2 x 1 cos 2x

FSc-II / Ex- 1.3 - 5

2

lim

x0

sin x

x

2

2

lim

x0

sin x

x

2

2(1)2

2

(vii)

Do yourself by rationalizing

(viii) lim sin2 lim sin sin

0

0

lim sin limsin (1) (0) 0

0

0

(x) lim sec x cos x

x0

x

1 cos x

1 cos2 x

lim cos x

lim cos x

x0

x

x0

x

lim1 cos2 x lim sin2 x lim sin x sin x

x0 x cos x

x0 x cos x

x0 x cos x

lim sin x lim sin x 1 sin(0) 1 0 0

x0 x x0 cos x

cos(0)

1

(xi) lim1 cos p x0 1 cos q

2sin2 p

lim

x0

2 sin 2

2 q

2

lim x0

sin 2

p 2

1 sin2 q

2

sin2 x 1 cos x 22

lim sin2 p

x0

2

p 2

2 p

2

1 q 2

2

sin2 q . 2

2 q

2

2

lim

x0

sin2 p

2 p 2

2

p 2

2

sin 2

1 q

q

2

2

.

q 2

2

2

lim x0

sin p 2

p

2

2

1

sin q 2

q

p2 2

2

q2

2

4

4

2

p2 q2

lim x0

sin p 2

p

2

2

lim

x0

1

sin q 2

q

2

2

p2 q2

(1)2

1 (1)2

p2 q2

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