Exercise 1.3 (Solutions)
Merging man and maths
Calculus and Analytic Geometry, MATHEMATICS 12
Available online @ , Version: 2.1.2
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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