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Calculus and Analytic Geometry, MATHEMATICS 12
Available online @ http://www.mathcity.org, Version: 2. 1. 2
Important Limits
I.
1 lim
n n n
x a
x a na x a
, where n is integer and a 0.
II.
0
lim x 2
x a a
x (^) a
III.
0
lim 1
n
n
e n
^
IV.
1 lim 1 x x
x e
V.
0
lim ln
x
x
a a x
, where a 0.
VI.
0
lim ln 1
x
x
e e x
VII. If is measured in radian, then
0
sin lim 1
Question # 1
(i) 3
lim(2 4) x
x
3 3
lim(2 ) lim(4) x x
x
3
2lim( ) 4 x
x
(ii)
2
1
lim 3 2 4 x
x x
2 3(1) 2(1) 4 3 2 4 5.
(iii)
2
3
lim 4 x
x x
2 (3) (3) 4 9 3 4 16 4.
(iv)
2
2
lim 4 x
x x
2 2 2 4 = 0.
(v)
3 2
2
lim 1 5 x
x x
3 2
2 2
lim 1 lim 5 x x
x x
3 2 (2) 1 (2) 5
(vi)
3
2
lim x 3 2
x x
x
3 2( 2) 5( 2)
Question # 2
(i)
3
1
lim x 1
x x
x
2
1
lim x 1
x x
x
^1
lim x 1
x x x
x
1
lim ( 1) x
x x
( 1)( 1^ ^ 1) 2
(ii)
3
0 2
lim x
x x
x x
2
0
lim x ( 1)
x x
x x
2
0
lim x 1
x
x
(iii)
3
2 2
lim x 6
x
x x
3 3
2 2
lim x 3 2 6
x
x x x
2
2
lim x ( 3) 2( 3)
x x x
x x x
2
2
lim x ( 3)( 2)
x x x
x x
2
2
lim x ( 3)
x x
x
2 (2) 2(2) 4)
(iv)
3 2
1 3
lim x
x x x
x x
3
1 2
lim x ( 1)
x
x x
3
1
lim x ( 1)( 1)
x
x x x
2
1
lim x ( 1)
x
x x
2
1
lim x (1)(1 1)
(v)
3 2
lim 1 2 x 1
x x
x
2
1
lim x ( 1)( 1)
x x
x x
2
1
lim x ( 1)
x
x
2 1
(vi)
2
4 3 2
lim x 4
x
x x
2
4 2
lim x ( 4)
x
x x
4 2
lim x ( 4)
x x
x x
4 2
lim x
x
x
2
(vii) 2
lim x 2
x
x
^2
lim x (^2 )
x x
x (^) x
^
2 2
2
lim 2 2
x
x
x x
2
lim 2 2
x
x
x x
2
lim x x 2
(viii) 0
lim h
x h x
h
0
lim h
x h x x h x
h (^) x h x
2 2
0
lim h
x h x
h x h x
0
lim h
x h x
h x h x
Now put 180
x t
(^) i.e.
180 t x
When x 0 then t 0 , so
0 0
sin (^) sin 180 lim lim x x 180
x t
x t
0
sin lim 180 x
t
t
by law of sine
(iii) 0
1 cos lim sin
0
1 cos 1 cos lim sin 1 cos
2
0
1 cos lim sin 1 cos
2
0
sin lim sin 1 cos
sin lim 1 cos
sin(0)
1 cos(0)
(iv)
sin lim x
x
x
Put t x x t
When x then t 0 , so
0
sin sin( ) lim lim x t
x t
x t
(^)
0
sin lim t
t
t
sin sin 2
t t
sin t
1 By law of sine.
(v) 0
sin lim x sin
ax
bx 0
limsin x sin
ax bx
0
limsin
sin
x
ax ax ax bx bx bx
^
0
sin 1 lim x sin
ax ax ax bx bx bx
0
0
sin 1 lim sin lim
x
x
a ax
b ax bx
bx
a
b
a
b
by law of sine
(vi) 0
lim x tan
x
x 0
lim sin
cos
x
x
x
x
0
lim cos x sin
x x x
0
lim cos x sin
x x
x
0
0
limcos sin lim
x
x
x x
x
(vii) 2 0
1 cos 2 lim x
x
x
2
0 2
2sin lim x
x
x
2 1 cos 2 sin 2
x x
2 2sin x 1 cos 2 x
2
0
sin 2lim x
x
x
2
0
sin 2 lim x
x
x
2 2(1) 2
(vii) Do yourself by rationalizing
(viii)
2
0
sin lim
sin lim sin
0 0
sin lim limsin
(x) 0
sec cos lim x
x x
x
0
cos cos lim x
x x x
2
0
1 cos
cos lim x
x
x
x
2
0
1 cos lim x cos
x
x x
2
0
sin lim x cos
x
x x
0
sin sin lim x cos
x x
x x
0 0
sin sin lim lim x x cos
x x
(^) x x
sin(0) 1 cos(0)
(xi) 0
1 cos lim x 1 cos
p
q
2
(^0 )
2sin 2 lim
2sin 2
x
p
q
2 1 cos sin 2 2
x x
2
(^0 )
lim sin 2 sin 2
x
p
q
2
2
0 2 2
2 2
2
2 2
2
lim sin 2
sin. 2
x
p
p q
q
p
q
2 2
0 2 2 2
2
2
2
2
2
sin 2 1 lim
sin (^2).
x
p
p
q
q
p
q
2 2 2
0 2 2 2
sin 2 1 4 lim
sin (^4) (^2 )
x
p p
p (^) q q
q
^
^
2
2
2 0 2
0
sin 2 1 lim
sin (^2 ) lim
x
x
p
p
q p q
q
^
^
2 2 2 2
p
q
2
2
p
q
0 1
ln lim
ln 1
y y
a
y
1
0
ln
limln 1 y y
a
y
ln ln
m x m x
1
0
ln
ln lim 1 y y
a
y
ln
ln
a
e
1
0
lim 1 x x
x e
ln
a ln a ln e 1
Question # 4
(i)
2 1 lim 1
n
n n
2 1 lim 1
n
n n
^
2 e
(ii)
lim 1
n
n n
1 2 1 lim 1
n
n n
^
1 2 e e
(iii)
lim 1
n
n n
1 1 lim 1
n
n (^) n
^
^
1 e
e
(iv)
lim 1 3
n
n n
3
1 3 lim 1 3
n
n n
1 (^3 ) 1 lim 1 3
n
n n
^
1
e^3
(v)
lim 1
n
n n
4 4
4 4 4 4 lim 1 lim 1
n n
n (^) n n n
^
4 e.
(vi)
2
0
lim 1 3 x x
x
6 3 0
lim 1 3 x x
x
1 6 3 0
lim 1 3 x x
x
6 e
(vii)
2
1 2
0
lim 1 2 x x
x
2
2 (^2 )
0
lim 1 2 x x
x
2
1 2 (^2 )
0
lim 1 2 x x
x
2 e
(viii)
1
0
lim 1 2 h h
h
2 2 0
lim 1 2 h h
h
1 2 2 0
lim 1 2 h h
h
2 e
2
e
(ix) lim 1
x
x
x
x
lim
x
x
x
x
lim
x
x
x
x x
lim 1
x
x (^) x
1 1 lim 1
x
x (^) x
^
1 e
e
(x)
1
0 1
lim
1
x
x (^) x
e
e
; x 0
Put x t where t 0
When x 0 then t 0 , so 1
0 1
lim
1
x
x (^) x
e
e
1
0 1
lim
1
t
t (^) t
e
e
1 0
1 0
e
e
e
e
e 0 e
(^)
1
(xi)
1
0 1
lim
1
x
x (^) x
e
e
; x 0
1
1
(^0 )
1
= lim 1 1
x
x
x x
x
e
e
e
e
1
0
1
= lim 1 1
x
x
x
e
e
1 0
1 0
e
e
e
e
Error Analyst
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Book: Exercise 1.3 (Page 27)
Calculus and Analytic Geometry Mathematic 12
Punjab Textbook Board, Lahore.
Edition: May 2013.
Made by: Atiq ur Rehman ([email protected])
Available online at http://www.MathCity.org in PDF Format (Picture format to view
online).
Page Setup used A4.
Printed: December 19, 2014.
