View
219
Download
0
Category
Preview:
Citation preview
8/14/2019 Limit and Continuty
1/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
2/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
At the end of the lesson, students are able to:
2) define limits intuitively
3) explain the concept of limits at anygiven
point asa
3) state and explain the basic properties oflimits
4) explain cases where
5) find the limit using substitution,factorizationand multiplicationof
conjugates
Lxfax
=
)(lim
0
0
)(lim
)(lim=
xg
xf
ax
ax
8/14/2019 Limit and Continuty
3/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
1) discuss one-sided limits
2) explain the existence of
3) define infinite limits4) define limits at infinity
Lxfax
=
)(lim
8/14/2019 Limit and Continuty
4/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
Limit?
8/14/2019 Limit and Continuty
5/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
Ive never heard
of that!
8/14/2019 Limit and Continuty
6/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
7/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
we are not interested in thevalue off(x) whenx= a.
we are interested in the behavior
off(x) asxcomes closer andcloser to a value ofa.
the notation; Lxfax = )(lim
8/14/2019 Limit and Continuty
8/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
)1(lim 20
+
xx
EXAMPLE:
Determine
21.251.06251.00000
1?
1.000001
1.0625
1.252f(x)=x2+
1
10.50.250.0010-0.001-0.25-0.5-1x
xapproaching 0 fromleft
xapproaching 0 from right
8/14/2019 Limit and Continuty
9/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
The graph of 1)( 2 += xxf
+
0x
0x
y=x2 + 1
y=f(x)
x
1
1)1(lim)1(lim2
0
2
0=+=+
+
xx
xx
0
8/14/2019 Limit and Continuty
10/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
11/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
The Diner Example
8/14/2019 Limit and Continuty
12/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
13/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
14/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
15/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
16/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
17/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( ) 2xxf =
8/14/2019 Limit and Continuty
18/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( ) 2xxf =
Different
values of xhave
differentvalues of y
8/14/2019 Limit and Continuty
19/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( ) 2xxf =
8/14/2019 Limit and Continuty
20/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( ) 2xxf =
8/14/2019 Limit and Continuty
21/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
4limlim 2
2
2
2
==+
xxxx
8/14/2019 Limit and Continuty
22/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( ) 4lim2=
xfx
8/14/2019 Limit and Continuty
23/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( ) 2862
+
= x
xx
xg
8/14/2019 Limit and Continuty
24/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( ) 2862
+
=x
xxxg
8/14/2019 Limit and Continuty
25/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
graph is discontinuousat (2, -2)
8/14/2019 Limit and Continuty
26/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( )
( )( )
( )
( )0
02
22
81242
22
82622
2
86
2
2
=
+=
+=
+=
g
g
g
x
xxxg
8/14/2019 Limit and Continuty
27/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( )
( )0
02
2
862
=
+=
g
x
xxxg
You cant divideany number by 0.How ugly is that?
8/14/2019 Limit and Continuty
28/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
( )2
862
+=
xxxxg
A limit still existsat x = 2
(even though
theres a holethere!)
8/14/2019 Limit and Continuty
29/92
8/14/2019 Limit and Continuty
30/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
31/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
32/92
8/14/2019 Limit and Continuty
33/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
34/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
35/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
36/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
37/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
38/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
39/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
40/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
41/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
42/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
43/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
exists if and only if)(lim xfax
)(lim)(lim xfxf axax + =
8/14/2019 Limit and Continuty
44/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
45/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
46/92
8/14/2019 Limit and Continuty
47/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
)(lim)(lim44
xfxfxx
+
8/14/2019 Limit and Continuty
48/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
exists
even though f(x)is discontinuous
at x = 2
)(lim2
xfx
8/14/2019 Limit and Continuty
49/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
50/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
51/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
1. Substitution
2. Factorization
3. The Conjugate Method
8/14/2019 Limit and Continuty
52/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
53/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
54/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
55/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
56/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
57/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
Factorization
8/14/2019 Limit and Continuty
58/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
59/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
60/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
61/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
62/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
63/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
64/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
65/92
8/14/2019 Limit and Continuty
66/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
67/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
68/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
69/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
70/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
71/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
72/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
INFINITE LIMITS
8/14/2019 Limit and Continuty
73/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
INFINITE LIMITS
8/14/2019 Limit and Continuty
74/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
75/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
76/92
8/14/2019 Limit and Continuty
77/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
f(x) is not defined at x = 3
8/14/2019 Limit and Continuty
78/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
79/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
80/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
81/92
8/14/2019 Limit and Continuty
82/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
83/92
8/14/2019 Limit and Continuty
84/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
LIMITS AT
INFINITY
8/14/2019 Limit and Continuty
85/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
86/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
87/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
88/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
89/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
90/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
91/92
9291908988878685848382818079787776757473727170
6968676665646362616059585756555453525150494847
4645444342414039383736353433323130292827262524
2322212019181716151413121110987654321
8/14/2019 Limit and Continuty
92/92
2322212019181716151413121110987654321
Recommended