Section 1'3 Limits of Functions and Continuity

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Section 1.3 - Limits of Functions and Continuity
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The Limit of a Function
The 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 (x1)
o
f(a) ? L
f (a) L
?
?
f (x2)
f (x2)
a
x1
x2
a
x1
x2
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The Limit of a Function
If the function values as x approaches a from
each side of a do not yield the same function
value, the function does not exist.
?
L2
?
L1
a
x
x
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The Limit of a Function
If 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.
f ? 8
?
?
a
x
x
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The Limit of a Function
If as x approaches infinity (or negative
infinity), f (x) approaches L, then the limit as
x approaches a of f (x) is L.
?
f (x)
L
x
x ? 8
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Evaluating Limits
Two methods of evaluating limits are (1) graphing
the function and/or (2) choosing values for the
input close a and on either side of a. If the,
after substituting these values into the
function, they appear to converge to a value,
this is usually the limit.
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Limits and Asymptotes
Limits can be used to determine vertical and
horizontal asymptotes. If
then the function has a vertical asymptote at x
a.
If
then the function has a horizontal asymptote at y
L.
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Continuity
A function is continuous over an interval of x
values if it has no breaks, gaps, nor vertical
asymptotes on that interval.
o
a
b
c
a
b
c
Not Continuous on (a, b) since discontinuous at x
c
Continuous on (a, b)
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Continuity
A function is continuous on an interval of x if
it is continuous at every value x c in the
interval. A function is continuous at x c if
the following is true and neither side equals DNE.
L1
o
L1
a
b
c
a
b
c
Not Continuous on (a, b) since discontinuous at x
c
Continuous on (a, b)
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Continuity Types
Is the following continuous?
L2
L1
c
a
b
c
a
b
Infinite Discontinuity
Jump Discontinuity
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Continuity Types
Is the function continuous?

L2
o
L1
a
b
c
Removable Discontinuity
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Examples
Page 46 6, 12
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Continuous Without Restrictions
Often we are given functions where the inputs can
be any real number. For example, the Fahrenheit
to Celsius formula has as its input a Fahrenheit
temperature any Fahrenheit temperature. This
could be 93? or 101.23?. When the input (domain)
value has no restrictions, we say the function is
continuous without restrictions or simply
continuous.
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Continuous With Discrete Interpretation
If, for example, we have a function that gives
the profit for a given number of units sold. We
often use a continuous function to model this
situation, but the input (domain) value can be
only discrete units 10, 20, 30, etc. We cannot
use input of 12.5 for example. When the input
(domain) value has this type of restriction yet
we use a continuous function to model the
situation anyway, we say the function is
continuous with a discrete interpretation.
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Examples
Page 46 18, 20, 26, 30
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Compound Interest Formula
where A is the amount after t years with a
principal (initial amount) P and an annual rate r
where the interested in compounded n times per
year.
Here, the interest formula is continuous, but the
interest is being computed at the end of each
compounding period. This makes it continuous with
a discrete interpretation.
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Continuous Compound Interest Formula
where A is the amount after t years with a
principal (initial amount) P and an annual rate r.
Here, the interest formula is continuous and the
interest is being computed at every moment in
time. This makes it continuous without
restrictions.
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Examples
Page 46 14