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Continuity, End Behavior, and Limits (vocabulary (discontinuous functions,…
Continuity, End Behavior, and Limits
objectives
Use limits to determine the
continuity of a function, and apply
the Intermediate Value Theorem to
continuous functions.
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vocabulary
The graph of a continuous function has no breaks,
holes, or gaps.
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discontinuous functions, or functions that are not continuous. Functions can have many different types of discontinuity.
A function has an infinite discontinuity
at x = c if the function value increases
or decreases indefinitely as x
approaches c from the left and right.
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A function has a jump discontinuity
at x = c if the limits of the function as
x approaches c from the left and right
exist but have two distinct values.
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function has a removable
discontinuity if the function is
continuous everywhere except for a
hole at x = c.
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examples
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2
Determine between which consecutive integers the real zeros of each function are located on
the given interval.
(x) = x 3 - 4x + 2; [-4, 4]
x= -4 f(x)= -46
x=-3 f(x)=-13
x=-2 f(x)= 2
x=-1 f(x)= 5
x=0 f(x)= 2
x=1 f(x)=-1
x=2 f(x)=2
x=3 f(x)=17
x=4 f(x)= 50
Because f (-3) is negative and f (-2) is positive, by the
Location Principle, f (x) has a zero between -3 and -2.
The value of f (x) also changes sign for 0 ≤ x ≤ 1 and
1 ≤ x ≤ 2. This indicates the existence of real zeros in
each of these intervals.
The graph of f (x) shown at the right supports the
conclusion that there are real zeros between -3 and
-2, 0 and 1, and 1 and 2.
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