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๐(โ4)= ๐(โ1)= ๐(1)= lim ๐ฅโ1 ๐ ๐ฅ = lim ๐ฅโ2 ๐(๐ฅ) = lim ๐ฅโโ3 ๐(๐ฅ) =
Warm-Up Find the indicated values based on the graph below: ๐(โ4)= ๐(โ1)= ๐(1)= lim ๐ฅโ1 ๐ ๐ฅ = lim ๐ฅโ2 ๐(๐ฅ) = lim ๐ฅโโ3 ๐(๐ฅ) = lim ๐ฅโ โ1 โ ๐ ๐ฅ = lim ๐ฅโ โ1 + ๐(๐ฅ) = lim ๐ฅโโ1 ๐(๐ฅ) =
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F(x) J(x) K(x) G(x) H(x) Warm-Up Given the graph of f(x),
which is f(|x|) graph? J(x) K(x) G(x) H(x)
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Warm-Up
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1-4: Continuity and One-Sided Limits
Objectives: Define and explore properties of continuity Discuss one-sided limits Introduce Intermediate Value Theorem ยฉ2002 Roy L. Gover (
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Definition f(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.
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Examples Continuous Functions
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Examples Discontinuous Functions
Infinite discontinuity (non-removable) Jump Discontinuity (non-removable) Removable discontinuity
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f(x) is continuous at x=c if and only if:
Definition f(x) is continuous at x=c if and only if: 1. f (c) is defined โฆand exists โฆand 3.
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Examples Discontinuous at x=2 because f(2) is not defined x=2
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Examples Discontinuous at x=2 because, although f(2) is defined, x=2
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Definition f(x) is continuous on the open interval (a,b) if and only if f(x) is continuous at every point in the interval.
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Try This Find the values of x (if any) where f is not continuous. Is the discontinuity removable? Continuous for all x
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Try This Find the values of x (if any) where f is not continuous. Is the discontinuity removable? Discontinuous at x=o, not removable
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Definition f(x) is continuous on the closed interval [a,b] iff it is continuous on (a,b) and continuous from the right at a and continuous from the left at b.
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f(x) is continuous from the right at a a
Example f(x) is continuous from the right at a f(x) is continuous on (a,b) a f(x) is continuous from the left at b f(x) b f(x) is continuous on [a,b]
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Definition is a limit from the right which means x๏ฎ c from values greater than c
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Definition is a limit from the left which means x๏ฎ c from values less than c
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Example Find the limit of f(x) as x approaches 1 from the right:
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Example Find the limit of f(x) as x approaches 1 from the left:
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Example Find the limit of f(x) as x approaches 1:
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Important Idea Theorem 1.10: exists iff
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Try This Use the graph to determine the limit, the limit from the right & the limit from the left as x๏ฎ0.
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Try This Use the graph to determine the limit, the limit from the right & the limit from the left as x๏ฎ1. x=1
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Intermediate Value Theorem
Theorem 1.13: If f is continuous on [a,b] and k is a number between f(a) & f(b), then there exists a number c between a & b such that f(c ) =k.
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Intermediate Value Theorem
f(a) k c f(b) b a
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Intermediate Value Theorem
an existence theorem; it guarantees a number exists but doesnโt give a method for finding the number. it says that a continuous function never takes on 2 values without taking on all the values between.
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Example Ryan was 20 inches long when born and 30 inches long when 9 months old. Since growth is continuous, there was a time between birth and 9 months when he was 25 inches long.
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Try This Use the Intermediate Value Theorem to show that
has a zero in the interval [-1,1].
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Solution therefore, by the Intermediate Value Theorem, there must be a f (c)=0 where
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Lesson Close 3 things must be true for a function to be continuous. What are they?
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Assignment Page 78 #1-6 all,7-19 odd,27-49 odd, 57, and 59
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