limits of functions and continuity. |a|a |x1|x1 |x2|x2 f (a) = l |a|a f(a) ≠ l o the limit of a...
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![Page 1: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/1.jpg)
Limits of Functions and Continuity
![Page 2: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/2.jpg)
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a
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x1
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x2
f (a) = L
lim ( )
( )
x af x L
f a exists
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a
f(a) ≠ L
lim ( )
( )
x af x L
f a does not exists
o
The Limit of a FunctionThe limit as x approaches a (x → a) of f (x) = L means that as x gets closer and closer to a (on either side of a), f (x) must approach L. Here, f (a) does not need to exist for the limit to exist.
f (x1)
f (x2)
f (x1)
f (x2)
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x1
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x2
![Page 3: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/3.jpg)
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L1
L2
The Limit of a FunctionIf the function values as x approaches a from each side of a do not yield the same function value, the function does not exist.
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2
lim ( )
lim ( )x a
x a
One Sided Limits
f x L
f x L
Since lim ( ) lim ( ), lim ( ) .x ax a x a
f x f x f x DNE
![Page 4: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/4.jpg)
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The Limit of a FunctionIf as x approaches a from either side of a, f (x) goes to either infinity or negative infinity, the limit as x approaches a of f (x) is positive or negative infinity respectively.
lim ( )
lim ( )x a
x a
One Sided Limits
f x
f x
Since lim ( ) lim ( ), then lim ( ) .x ax a x a
f x f x f x
f → ∞
![Page 5: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/5.jpg)
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x
The Limit of a FunctionIf as x approaches infinity (or negative infinity), f (x) approaches L, then the limit as x approaches a of f (x) is L.
lim ( )x
f x L
L
f (x1)
x → ∞
![Page 6: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/6.jpg)
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b
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ContinuityA function is continuous over an interval of x values if it has no breaks, gaps, nor vertical asymptotes on that interval.
Continuous on (a, b)
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c
Not Continuous on (a, b) since discontinuous at x = c
![Page 7: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/7.jpg)
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ContinuityIn other words, a function is continuous at x = c if the following is true.
Continuous on (a, b)
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Not Continuous on (a, b) since discontinuous at x = c
)()(lim cfxfcx
![Page 8: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/8.jpg)
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c
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a
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Continuity Types
Is the following continuous?
Infinite Discontinuity
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c
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a
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Jump Discontinuity
![Page 9: Limits of Functions and Continuity. |a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x)](https://reader035.vdocuments.us/reader035/viewer/2022072006/56649f555503460f94c78d0b/html5/thumbnails/9.jpg)
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a
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Continuity TypesIs the function continuous?
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c
Removable Discontinuity
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