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Unformatted text preview: Project 9
MAC 2233
1. Consider the following piecewise deﬁned function for various choices of constants p and q .
Sketch its graph in the case that p = q = 1. −p
x < −1 x 2
x = −1
f (x) = 2−x
−1 < x < 2 √ x+q
x≥2 Determine whether f (x) is continuous at x = −1, x = 0, and x = 2. Classify each discontinuity as inﬁnite, jump, or removable.
jump discontinuities at x = −1, 2
continuous at x = 0
2. For what choices of p and q do the limits lim f (x) and lim f (x) both exist? Do these
x→−1 x→2 choices make f (x) continuous? Why or why not? Sketch the graph of f (x) with this p and q . p = 3 and q = −2 If the function f (x) is now still not continuous at some point, can it be redeﬁned to make it
continuous at that point, without altering the behavior of f (x) near the point? If so, how?
What do we call this type of discontinuity?
f (−1) = 3 removes the ”hole” at x = −1
Evaluate the limits at inﬁnity for the function f (x).
limit as x → ∞ is ∞
limit as x → −∞ is 0 ...
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This note was uploaded on 12/27/2011 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.
 Spring '08
 Smith
 Calculus

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