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Continuity, End Behavior, and Limits 1-3 (Vocabulary (discontinuous…
Continuity, End Behavior, and Limits 1-3
Learning goals
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. You can trace the graph of a continuous
function without lifting your pencil.
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discontinuous functions, or functions that are not continuous.
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.
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.
A function has a removable
discontinuity if the function is
continuous everywhere except for a
hole at x = c.
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Examples
Q. Determine whether the function f (x) = x 2 + 2x – 3 is continuous at x = 1. Justify using the continuity test.
Answer. continuous, lim f(x) = f(1)
Q. Determine whether the function f (x)= 1/x^2 - 1
is continuous at x = 1. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.
Answer. infinite discontinuity
Q. Determine between which consecutive integers the real zeros of f (x) = x^3 + 2x + 5 are located on the interval [–2, 2].
Answer. –2 < x < –1.
Q. Use the graph of
f (x) = x 3 + x 2 – 2x + 1 to describe its end behavior. Support the conjecture numerically.
Answer.
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