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1.4 One-Sided Limits and Continuity
Definition
A function is continuous at c if the following three conditions are met
2. Limit of f(x) exists
1. f(c) is defined
3. Limit of f(x) is cc
Definition
If a function is defined on an interval I, except at c, then the function is said to have a discontinuity at c such as a hole, break or asymptote
One-Sided Limits
Approach a function from different directions both graphically and analytically
1)Limits from the right
2)Limits from the left
limx c
f (x)
limx c
f (x)
Existence of a Limit
• Let be a function and let c and L be real numbers. The limit of as x approaches c is L if and only if (iff)
f
f (x)
limx c
f (x) limx c
f (x) L
Consider
( | )a c b
( | )a c b
( | )a c b
f (c) is undefined
)(lim)(lim xfxfcxcx
limx cf (x) f (c)
limx c
f (x) limx c
f (x)
1) Find
limx 0
x x
x
limx 0
x x1
2
x1
2
limx 0
x1
2 x1
2 1
x1
2
1
21
21
21
)0(
1)0()0(
limx1f (x), if f (x)
x 3 1, x < 1
x +1, x 1
Left Right
2) Find
2) Find
limx1f (x), if f (x)
x 3 1, x < 1
x +1, x 1
limx1
x 3 1
Left Right
2
limx1
x 1
2
limx1f (x) 2
By existence theorem
f (x) x 2
x 2
2 x,
2
)2(
2 x,2
)2(
x
xx
x
3) Determine if the limit exists at x = -2 if
3) Determine if the limit exists at x = -2 if
f (x) x 2
x 2
limx 2
(x 2)
x 2
Left Right
1
limx 2
(x 2)
x 2
1
limx 2
f (x) DNE
(x 2)
x 2, x 2
(x 2)
x 2, x > 2
Continuity at a point
• Let f(x) be defined on an open interval containing c, f(x) is continuous at c if
a. is defined (exists)
b. exists (one-sided limits are equal)
c. The
f (c)
limx cf (x)
limx cf (x) f (c)
Discontinuity
Removable: the function can be redefined (hole discontinuity)
Discontinuity
Non - Removable: a. Jump - breaks at a particular value
b. Infinite discontinuity - vertical asymptote
4) Find the x - values where is not continuous and classify
f (x) x, x < 1
3, x 1
2x 1, x 1
f (1)a. exists
b.
limx1
f (x) limx1
(2x 1)
1
limx1
f (x) limx1
x
1
3
c.
limx1f (x) f (c)
Removable Point Discontinuity
5) Find the x - values at which is not continuous,
is the discontinuity removable?
f (x) x
x 2 1
)1(1)(
xx
xxf
(x 1)(x 1) 0
x 1,x 1
Non-removable: asymptotes
6) Find the x - values at which
is not continuous,
is the discontinuity removable?
f (x) x 3
x 2 9
f (x) x 3
(x 3)(x 3)
f (x) 1
(x 3)
Non-removable
x 3
Removable
x 3
7) If is continuous at ,
then
f (x) x 2 9
x 3
x 3
f ( 3)
f (x) (x 3)(x 3)
x 3
f ( 3) 3 3
f ( 3) 6
HOMEWORK
• Page 79 # 1-11, 18, 19, and 20