Download sect. 1-4 continued

Sect.1.4 continued
One-Sided Limits and
Continuity
Properties of Continuity
If b is a real number and f (x) and g( x) are both
continuous at x = c then, the following are also
continuous
1. Scalar:
b f
2. Sum/Difference:
3. Product:
f g
f g
f
4. Quotient:
g
5. Composite: If g(x) is continuous at c and f(x)
at g(c), then f[g(x)] is also continuous at c
 x 2  2 x,
8) Is f ( x)   3
 x  6 x,
x  2
x  2
continuous at
x = –2
Check the three conditions
a. f (2) exists
3
lim
f
(
x
)

lim
(
x
 6 x)
b.


x2
x2
4
lim  f ( x)  lim  ( x 2  2 x)
x 2
x 2
0
Jump Discontinuity
3x  5, x  1
9) Determine if f (x)  
is
x 1
2,
continuous for x = 1
Check the three conditions
a.

f (1)
b.
lim f (x)  lim (3x  5)
x 1
exists
x 1
 2
c. lim f (x)  f (1)
x 1

Limit DNE
ax  5,
10) For what value of ‘a’ is f ( x)   2
x  x,

continuous
Check the three conditions
a. f ( 4) exists
2
lim
f
(
x
)

lim
(
x
 x)
b.


x4
x4
 12
lim f ( x)  lim (ax  5)
x4
x4
 4a  5
12  4a  5
7  4a
7
a
4
x4
x4
x  1
2,

f ( x)  ax  b, - 1  x  3
11)
is continuous for
 2,
x3

all real numbers for what value of ‘a’ and ‘b’
x  1
x 3
a. f (1) exists
a. f (3) exists
b. lim  f ( x)  lim  (ax  b)
b.
x  1
x  1
lim f ( x)  lim (ax  b)
x 3
 3a  b
 a  b
lim  f ( x)  lim  (2)
x  1
x  1
2
2  a  b
x 3
lim f ( x)  lim (2)
x 3
x 3
 2
 2  3a  b
12)
x3  1
f ( x)  
x  1
x 1
x 1
is it continuous?
x 1
a. f (1) exists
c. f ( x)  lim f ( x)
x1
b.
lim f ( x)  lim f ( x)
x 1
x 1
HOMEWORK
• Page 80 # 35-53 odd, 63-66 all