Download Example 1 Find the points at which the function is not continuous

CP Calculus
Block 19
HW-Review p88,
84;90;96;108;114;
120;126;
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CP Calculus
Block 19
HW p98,x3’s
3-48
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Ch 2.4 Continuity and
One-sided Limit
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Do Now:
x 2
,
x

0
 x  1
f( x )  
1

,
x

0
2
 x
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x 2
1
1
x 1
x 1
2
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1
2
x
5
2
0
lim f ( x )
x 
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2
1
lim f ( x )
x 
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lim f ( x )
x 0
2
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
8
lim f ( x )
x 0
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2
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Objective 1:
Determine the
continuity of the
functions
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Definition A:
Continuity at an
interior point
lim f(x) = f(c)
x c
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Definition B:
Continuity on
closed interval [a,b],
left endpoint a or
right endpoint b
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lim+ f(x) = f(a)
x a
lim- f(x) = f(b)
x b
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Definition C: A
function is
continuous if it is
continuous at each
point of its domain
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The Continuity Test
The function y=f(x) is
continuous at x=c if
and only if all of the
following are true:
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i) f(c) exitsts
ii )limf(x) exists
x c
iii) lim f(x) = f(c)
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x c
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Existence of Limit
lim f ( x )  lim f ( x )
x c
x c
L
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Properties of
Continuity I
Scalar Multiple:
bf(x)
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Properties of
Continuity ii
Sum or difference
f + g, or f - g
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Properties of
Continuity iii
Product
f(x)g(x)
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Properties of
Continuity iv
Quotient
f(x)/g(x) g( x )  0
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Continuity of
Composite
Functions
f(g(x)) or g(f(x))
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Theorem: If f is
continuous at c and g
is continuous at f©,
then the composite
gof is continuous at c.
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Example 0 Given f(x)
1, x  0


2
f( x )   1  x , 0  x 1
 x  1, x  1

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1, x  0


f( x )   1  x2 , 0  x  1
 x  1, x  1

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a) Determine a
complete graph of
f(x)
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
b) Is f(x) continuous?

Explain.
f( x )  


Yes. At all points
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1, x  0
1  x2 , 0  x  1
x  1, x  1
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Example 1 Find the
points at which the
function is notx 
continuous
1
f( x ) 
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2
(x 2)
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2
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Example 2 Find the
points at which the
function is not
x


2,5
continuous
x 3
f( x )  2
x  3x  10
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Example 3 Find the
points at which the
1
function is not
f
(
x
)

continuous
2
x 1
No discontinuities
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Example 4 Find the
points at which the
function is not
continuous
f ( x )  2x  3
No discontinuities
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Example 5 Find the
points at which the
function is not
continuous
x=0
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x
f( x ) 
x
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Example 6 Find the
points at which the
function is not
x<1/3
continuous
f ( x )  3x  1
4
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Example 7 Find the
points at which the
function is not
No discontinuities
continuous
5
f( x )  2  x
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Example 8 Define
g(x) so that g(3)=6
2
g(x)=(x -9)/(x-3) is
continuous at x=3
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Example 9 Define
3
f(1) so that f (1 ) 
2
3
2
f(x)=(x -1)/(x -1) is
continuous at x=1
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Example 10 Given g(x)
1
 3
x
,
x


2
g( x )  
1
2
bx , x 

2
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What value should
be assigned to b to
make g(x)
continuous at
x=1/2?
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1
b
2
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Example 11: Find
the limit
0
limtan x
x 0
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Example 12: Find
the limit
lim sin(
x 0
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1

2
cos(tan x ))
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Example 13: Find the
vertical asymptote(s) of
3
f(x) =
(x + 2)
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Example 14: Find
the vertical
asymptote(s) of
1
f(x) = 2
x -x-2
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x=2
x = -1
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1
f(x) = 2
x - x -2
1
=
(x + 1)(x - 2)
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Example 15: Find the
end behavior
asymptote(s) of
x +1
f(x) = 2
x - x -6
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vertical asymptotes
end behavior asymptote (horizontal)
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Example 16
Find limf(x) if
x 
a )2
b )2
2x  7
f(x) = 3
2
x  x  x 7
3
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Example 17
Find limf(x) if
x 
3x  7
f(x) = 2
x 2
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a )0
b )0
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Example 18
Find limf(x) if
x 
f(x) =
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x
4
a )
b
)


x 1
3
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Example 19
Find limf(x) if
x 
a )0
b )0
10 x  x  31
f(x) =
6
x
5
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Example 20
Find limf(x) if
x 
a ) 1
b
)

1
4
x
f(x) = 4
3
2
x 7 x 7 x  9
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Lesson quiz 1: Given
 x  1, 1  x  0

2
x
,
0

x

1

f ( x )   1,
x 1
2 x  4, 1  x  2

2x 3
 0,
2
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 x  1, 1  x  0

2
x
,
0

x

1

f ( x )   1,
x 1
2 x  4, 1  x  2

2x 3
 0,
2
a) Does f(1) exist?
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yes,f (1)  1
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 x  1, 1  x  0

2
x
,
0

x

1

f ( x )   1,
x 1
2 x  4, 1  x  2

2x 3
 0,
2
b) lim f(x) exist?
x 1 yes,limf ( x )  2
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x 1
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 x  1, 1  x  0

2
x
,
0

x

1

f ( x )   1,
x 1
2 x  4, 1  x  2

2x 3
 0,
2
c) Does lim f(x) = f(1)?
x 1
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No
53
 x  1, 1  x  0

2
x
,
0

x

1

f ( x )   1,
x 1
2 x  4, 1  x  2

2x 3
 0,
2
d) Is f(x) continuous at x = 1?
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No
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 x  1, 1  x  0

2
x
,
0

x

1

f ( x )   1,
x 1
2 x  4, 1  x  2

2x 3
 0,
e) At what values of x
is f(x) continuous? all points in [-1, 3]
2
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except x = 0, 1, 2
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 x  1, 1  x  0

2
x
,
0

x

1

f ( x )   1,
x 1
2 x  4, 1  x  2
f) How should h be

0,
2

x

3


defined to make h a
h(
x
)

f
(
x
)
continuous extension
of f to the point x = 1? x  1,h(1)  2
2
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