Mathematics

Problem

Let

k

be a real–valued function defined on

(-\infty,\infty)

that satisfies the following limit conditions: •

\displaystyle \lim_{x\to -3} k(x)=4

•

\displaystyle \lim_{x\to 2^-} k(x)=1

•

\displaystyle \lim_{x\to 2^+} k(x)=5

•

k(2)=3

•

\displaystyle \lim_{x\to 6} k(x)=\infty

Your responses should reference these conditions explicitly. (a) Without calculating any algebraic expressions, describe in words what must happen to the values of

k(x)

x

approaches

-3

2

, and

6

. (b) Classify the type of discontinuity (if any) that must occur at

x=-3

x=2

, and

x=6

, providing justification for each classification. (c) Sketch a possible graph of

y=k(x)

on the interval

[-5,8]

that is consistent with all of the given limit conditions. Clearly label all relevant open or closed points and asymptotes. (d) Explain how your sketch from part (c) satisfies each of the stated limit conditions.

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