2.4 Limits and Continuity

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2.4 Continuity and its
Consequences
Thurs Sept 17
Do Now
Find the errors in the following
and explain why it’s wrong:
HW Review: p.80 #5 13 19 27
29 33 35
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•
•
•
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•
5) 1/2
13) -2/5
19) 1/5
27) 3
29) 1/16
33) let f(x) = 1/x and g(x) = -1/x
35) proof
Continuity - What does it
mean?
A function is said to be continuous on
an interval if its graph on that interval
can be drawn without interruption, or
without lifting your pencil.
• Holes and asymptotes are examples of
discontinuous functions
Definition of continuous
•
•
•
•
•
A function f is continuous at x = a when
1) f(a) is defined
2)
exists
3)
Otherwise, f is said to be discontinuous
at x = a
One-Sided Continuity
• A function f(x) is called:
– Left-continuous at x = c if
– Right-continuous at x = c if
What kind of functions are
continuous?
•
•
•
•
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Polynomials
Radical Functions on their domains
Sin x and cos x
Exponential functions
Logarithmic functions on their domains
Rational functions on their domains
Piecewise Functions
• These kind of functions are the big AP
type of problems
More Continuous Functions
• Thm- Suppose that f and g are
continuous at x = c. Then:
– 1) kf(x) for any constant k
– 2) ( f ± g) is continuous at x = c
– 3) ( f × g) is continuous at x = c
– 4) ( f ) is continuous at x = c if g(c) ¹ 0
g
and discontinuous if g(c) = 0
More Continuous Functions
• Thm- If f(x) is continuous on an interval I
with range R and its inverse exists, then
its inverse is continuous with domain R
Composite Functions
• If g(x) is continuous at x = c, and f(x) is
continuous at x = g(c), then f(g(x)) is
also continuous at x = c
3 Types of Discontinuities
• Removable Discontinuity
– Limit exists
– F(x) is not equal to the limit
– Can redefine function at discontinuity
• Jump Discontinuity
– Left and right side limits do not agree
– Cannot redefine
• Infinite Discontinuity
– One or both of each sided limits is infinite
Closure
• Journal Entry: What must be true for a
function to be continuous? What is an
example of a discontinuity? Which are
removable or not?
• HW: p.88-89 #1, 3-5, 17-33 odds, 55 57
59 63 65
Continuity Cont’d
Fri Sept 18
•
•
Do Now
Is the function f (x) = 16 - x 2
continuous at the following points?
1) X = 3
2) X = 4
HW Review: p.88-89 #1, 3-5,
17-33 odds, 55 57 59 63 65
1) [email protected] [email protected] [email protected]
LC
27) x = 2, jump,
3) X = 3 redefine g(3) = 4 29) t = (2n+1)pi/4, n = int
4) C = 1, redefine g(1) = 3 31) continuous for all
5) Omgicantfitthishere
33) x = 0, inf, neither
17) X = 0, inf, neither
55) show right lim = left
19) X = 1, inf, neither
57) c = 5/3
21) Even ints, jump, RC
59) a = 2, b = 1
23) X = 1/2, inf, neither
63) graph
25) Continuous for all x
65) graph
Classwork
• Side 1(p.53) #3, 4
• Side 2(p.153) #21 22 23 24 25
Closure
• Exit pass: Find all discontinuities of
x -1
f (x) = 2
x -1
For each discontinuity, state the type,
whether it is left/right continuous, and if
removable, redefine it so it is continuous
HW: none or finish worksheet
2.3-2.5 quiz soon
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