Continuity

Suppose that f (a) = g(a) = 0 that f ?(a) and g?(a) exist , and that g?(a) ? 0
Then
( )
( )
( )
lim ( )
f a
f a
g x
f x
x a ?
?
=
? [ H.W. ( prove it )
and
( )
lim ( )
( )
lim ( )
f x
f x
g x
f x
x a ?
?
=
?
Find the following limits by using L Hopital Rule :
Find The Following Limits ( By using L Hopital s Rule )
1-
?
?
=
? ?
? ?
=
?
?
?? 2
1
3 2
lim 2 2
x
x
x
meaning less
3
2
3
lim 2 =
x??
2-
0
0
5(0) 2(0)
2(0) 3(0)
5 2
lim 2 3 2
2
2
2
0
=
?
+
=
?
+
? x x
x x
x
meaning less
2
3
10(0) 2
4(0) 3
10 2
lim 4 3
0
?
=
?
+
=
?
+
?
? x
x
x
3-
0
0
0
lim 1 1 1 0 1
0
=
+ ?
=
+ ?
? x
x
x
meaning less
2
(1 0) 1
2
1
10 2
lim 1 2(1 ) 1 2
1 2
0
= + =
?
+
? ?
?
? x
x
x
4-
0
0
1 1
2 2
(1) (1) (1) 1
(1) 3(1) 2
1
lim 3 2 3 2
3
3 2
3
1
=
? +
? +
=
? ? +
? +
=
? ? +
? +
? x x x
x x
x
meaning less
2
3
4
6
6 2
lim 6
0
0
3 2 1
3 3
3 2 1
lim 3 3
2 1
2
1
= =
?
= ?
? ?
?
=
? ?
?
? ? x
x
x x
x
x x
جامعة بابل / كلية التربية للعلوم الصرفة / قسم الرياضيات 11 ( رياضيات ) 20
5-
0
0
0
1 1
cos( 2)
1 sin( 2)
cos( )
lim 1 sin( )
cos( )
sin( )
cos( )
lim [sec( ) tan( )] lim 1
2 2 2
=
?
=
?
=
?
? ?
??
?
? ??
?
? = ?
? ? ? ?
?
? ? ? x
x
x
x
x
x x
x x x
0
1
0
sin( 2)
cos( 2)
sin( )
lim cos( )
2
= = =
?
?
?
? ?
?
? x
x
x
6-
0
0
0
lim sin( ) sin(0)
0
= =
? x
x
x
meaning less 1
1
1
1
cos(0)
1
lim cos( )
0
? = = =
?
x
x
7-
0
0
0
ln(1) 0
(0)
lim ln( 1) 2 ln(0 1) 2(0) 0 2 2
=
?
=
+ ?
=
+ ?
? x
x x
x
meaning less = ?
?
=
?
= +
?
+
? 0
1
2(0)
2
0 1
1
2
2
( 1)
1
lim
0 x
x
x
8-
0
0
1 1
1 1
1 cos(0)
0 1
1 cos( )
lim 2 1
2 0
0
=
?
?
=
?
? ?
=
?
? ?
?
e
x
e x x
x
meaning less
0
0
0
2 2
sin(0)
2 2
sin( )
lim 2 2
2 0
0
=
?
=
?
=
?
?
?
e
x
e x
x
meaning less 4
1
4
cos(0)
4
cos( )
lim 4
2 0
0
? = = =
?
e
x
e x
x
9-
0
0
sin(0)
sin(0)
sin(3 )
lim sin(2 )
0
= =
? x
x
x
meaning less
3
2
3cos(0)
2cos(0)
3cos(3 )
lim 2cos(2 )
0
= =
? x
x
x
H.W : Find the following limits :
4
lim 2 0 2 ?
?
? x
x
x 3 8
lim 6 5
?
+
?? t
t
t cos( )
lim 2
2 x
x
x
?
?
?
? 3 2
lim 2 2
2
?
?
?? t
t t
t
3
lim cos( ) 1 2
? 3 ??
?
? x
x
x ? ?
?
??? ?
lim sin( )
4 3
lim 1 3
3
1 ? ?
?
? x x
x
x
0 2
lim cos( ) 1
t
t
t
?
?
1.73205
0.5
0.866025
2
1
2
3
cos(60)
4 60 tan ( ) tan(60) sin(60)
) 1
2
3 sin(90) sin(
2
) 1
6
2 sin(30) sin(
0
2
1 1
2
1 sinh(0)
1
0 0
? = ? = ? = = = =
? = =
? = =
=
?
=
?
? =
?
?
? ? ?
?
?
e e
جامعة بابل / كلية التربية للعلوم الصرفة / قسم الرياضيات 11 ( رياضيات ) 20
1 Show That
sinh(x) + cosh(x) = ex
1 tan( ) tan( )
tan( ) tan( ) tan( )
a b
a b a b
?
+
+ = ) cos( )
2
sin(a ? = ? x
?
2 Derivative
y4 + tan(xy) + sin(x) = 2 tan[sec3 ( 2 sin(2 ))] 2
?
y = x + x
sinh2 [sec 1(2 )].cos 1(1)
x
y = ? x ? y = [ln(x2 + sin?1(2x)]3
y = sin4[ln(x)]tan?1(x2 +1)
y = e2sin(3x) .csc?1(x2 + 2x +1)
sin (3 1) 2 1
.
1
2 + + ?
y = x ex
y = tanh[x2 + sin(2x)]
y = tan?1[sin(x)].tanh3 (e2x )
y = sinh2 [sin3 (x2 + 2x + 3)]
2 ( 1) tan ( )
2
ln( 1),sec( 2).
2 1
2 1
x x
x
x e
y x x e ?
+
+
+ +
=
y = sec h?2 [sec h(x2 +1)]sin?1[x2 + csc h(x)]
4
1
2 ?
=
x