Limit and Continuty
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8/14/2019 Limit and Continuty
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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
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1) discuss one-sided limits
2) explain the existence of
3) define infinite limits4) define limits at infinity
Lxfax
=
)(lim
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Limit?
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Ive never heard
of that!
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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
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)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
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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
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The Diner Example
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( ) 2xxf =
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( ) 2xxf =
Different
values of xhave
differentvalues of y
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( ) 2xxf =
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( ) 2xxf =
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4limlim 2
2
2
2
==+
xxxx
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( ) 4lim2=
xfx
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( ) 2862
+
= x
xx
xg
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( ) 2862
+
=x
xxxg
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graph is discontinuousat (2, -2)
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( )
( )( )
( )
( )0
02
22
81242
22
82622
2
86
2
2
=
+=
+=
+=
g
g
g
x
xxxg
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( )
( )0
02
2
862
=
+=
g
x
xxxg
You cant divideany number by 0.How ugly is that?
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( )2
862
+=
xxxxg
A limit still existsat x = 2
(even though
theres a holethere!)
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exists if and only if)(lim xfax
)(lim)(lim xfxf axax + =
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)(lim)(lim44
xfxfxx
+
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exists
even though f(x)is discontinuous
at x = 2
)(lim2
xfx
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1. Substitution
2. Factorization
3. The Conjugate Method
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Factorization
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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
