Ex 1 4 FSC Part2
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7/27/2019 Ex 1 4 FSC Part2
1/4
MathCity.orgMerging man and maths
Exercise 1.4 (Solutions)Calculus and Analytic Geometry,
MATHEMATICS 12
Available online @ http://www.mathcity.org, Version:
1.0.0Question # 1:
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )1
2
22
1 1
22
1 1
11 1
2 5 1
lim lim 2 5 2 1 1 5 2 1 5 2
lim lim 2 5 2 1 1 5 2 1 5 2
lim lim 2 lim 2
x x
x x
xx x
i f x x x c
f x x x
f x x x
f x f x f x
+ +
+
= + =
= + = + = + =
= + = + = + =
= = =
( ) ( )
( )( )
( )
( )
( )( )
( )
( )
( ) ( ) ( )
2
2 22
3
3 3
3
2 22
3
3 3
3
33 3
93
3
lim 9 3 99 9 9 0lim lim 0
3 lim 3 3 3 6 6
lim 9 3 99 9 9 0lim lim 0
3 lim 3 3 3 6 6
lim lim 0 lim 0
x
x x
x
x
x x
x
xx x
xii f x C
x
xxf x
x x
xxNow f x
x x
f x f x f x
i
+
+ +
+
+
= =
= = = = = =
= = = = = =
= = =
( ) ( )
( ) ( )5 5
5 5
lim lim 5 5 5
( 5) ( 5)
5
x x
ii f x x C
f x x x x
x x
= =
= =
+
+
( ) ( ) ( )
( ) ( )
55
5 5 5
lim 5 lim 5 5 5 0
lim lim 5 lim 5 5 5 0
xx
x x x
x x
f x x x
+ + +
= = = =
= = = =
( ) ( )
( )
5 5
5
lim lim 0
lim 0
x x
x
f x f x
f x
+
= =
=
Question # 2:
( )Discuss the continuity of f x at x c=
( ) ( )
( )
2 5 2
4 1 2
2
2
x if x
i f x x if x
c
Wehavetodiscussthecontinuityof f x at x
+
= + > =
=
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
2
22
22
2
2 2 2 5 4 5 9 .................... 1
lim ?
2 5 4 1
2
lim lim 2 5 2 2 5 4 5 9
lim lim 4 1 4 2 1 8 1 9
lim 9 ............... 2
x
xx
xx
x
a f
b f x
f x x f x x
f x x
and f x x
f x
+
= + = + =
=
= + = +
+
= + = + = + =
= + = + = + =
=
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FSc-II/Ex 1.4-2
( ) ( ) ( )
( ) ( )
( )
2
1 2
lim 2
2
x
c from and we get
f x f
f x is continous at x
=
=
( ) ( )
( ) ( )
( )
( )
( )
3 1 1
4 1 2
2 12 2
3 1 1
4 1
2 1
1
x if x
ii f x x c
x xif c f c f
is not defined so given function is discontinous
ii Correction
x if x
f x if x
x if x
c correction
= =
=
( )
( ) ( )
( ) ( ) ( )
1 41
3 1 2
1 4
f
f x x f x x
a f given
= +
= =
=
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )( ) ( )
1
11
11
1 1
1
lim ?
lim lim 3 1 3 1 1 2
lim lim 2 2 1 2
lim lim 2lim 2 ............................. 2
x
xx
xx
x x
x
b f x
f x x
and f x x
f x f xf x
+
+
=
= = =
= = =
= =
=
( ) ( ) ( )
( ) ( )
( )
1
1 2
lim 1
1
x
c From and we get
f x f
f x is discontinous at x
=
( ) ( )
3 1 1
1
2 1
x if x
iii f x c
x if x
( ) ( )
( )
1
1
a f is not defined
f x is discontinous at x =
Question # 3:
( )
( ) ( ) ( )
2
2
3 2
1 2 2
3 2
3 2 1 2 3
Giventhat
x if x
f x x if x
if x
f x x f x x f x
= <
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FSc-II/Ex 1.4-3
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( )
2
22
22
22
2 2
2
lim ?
lim lim 1 2 1 4 1 3
lim lim 3 3
lim lim 3
lim 3 ....................... 2
x
xx
xx
x x
x
b f x
f x x
f x
f x f x
f x
+
+
=
= = = =
= =
= =
=
( ) ( ) ( )
( ) ( )
( )
2
1 2 ,
lim 2
2
x
c From and we get
f x f
f x is continuous at x
=
=
( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
2
22
22
22
22 2
2
2 3 2 6 ............................ 1
lim ?
lim lim 3 3 2 6
lim lim 1 2 1 4 1 3
lim lim lim
2
x
xx
xx
xx x
ii At x
a f
b f x
f x x
and f x x
f x f x f x does not exist
f x is discontinuous at x
+
+
=
= =
=
= = =
= = = =
=
Question # 4:
( )
( ) ( )
2 1
2 1
?
2 1 2
Giventhat
x xf x
c x
c
f x x f x c
+ =
+ >
= +
= + = +
( )
( ) ( )
( ) ( )
1
1 1
1 1
lim
lim lim
lim 2 lim 2
1 2 2
1 2
1 2 1
x
x x
x x
f x exists
f x f x
x c
c
c
c c
+
=
+ = +
+ = +
= +
= =
Q
Question # 5:
( )
( ) ( )
( )
3
3
2 9 3
3
i
mx if x
f x n if x
x if x
here f n given
=
( )
( ) ( ) ( )
( ) ( )
3 3
3 3
3
lim lim 3
lim lim 2 9
x x
x x
f x is continuous at x
f x f x f
mx x n
+
=
= =
= + =
Q
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4/4
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FSc-II/Ex 1.4-4
( )( ) ( )3 2 3 9
3 6 9
3 3
3 3 , 3
1 , 3
m n
m n
m n
m n
m n
= + =
= + =
= =
= =
= =
( ) ( )
( ) ( )
2
2
4
4
4 4 16
mx if x
ii f x x if x
here f
