1.4 One-Sided Limits and Continuity. Definition A function is continuous at c if the following three...
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Transcript of 1.4 One-Sided Limits and Continuity. Definition A function is continuous at c if the following three...
![Page 1: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/1.jpg)
1.4 One-Sided Limits and Continuity
![Page 2: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/2.jpg)
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
![Page 3: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/3.jpg)
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
![Page 4: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/4.jpg)
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)
![Page 5: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/5.jpg)
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
![Page 6: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/6.jpg)
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)
![Page 7: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/7.jpg)
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(
![Page 8: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/8.jpg)
limx1f (x), if f (x)
x 3 1, x < 1
x +1, x 1
Left Right
2) Find
![Page 9: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/9.jpg)
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
![Page 10: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/10.jpg)
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
![Page 11: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/11.jpg)
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
![Page 12: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/12.jpg)
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)
![Page 13: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/13.jpg)
Discontinuity
Removable: the function can be redefined (hole discontinuity)
![Page 14: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/14.jpg)
Discontinuity
Non - Removable: a. Jump - breaks at a particular value
b. Infinite discontinuity - vertical asymptote
![Page 15: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/15.jpg)
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
![Page 16: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/16.jpg)
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
![Page 17: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/17.jpg)
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
![Page 18: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/18.jpg)
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
![Page 19: 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)](https://reader036.fdocuments.us/reader036/viewer/2022062313/56649f555503460f94c78d12/html5/thumbnails/19.jpg)
HOMEWORK
• Page 79 # 1-11, 18, 19, and 20