The function is represented graphically for

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

The function

r

is represented graphically for

-3\le x\le3

. Key features near

x=2

are summarized below. • For

x<2

, the curve approaches the line

y=\dfrac{1}{x-2}

. • For

x>2

, the curve follows

y=\dfrac{\sin\bigl(\pi(x-2)\bigr)}{x-2}

. • There is no point plotted at

x=2

. Respond to the following regarding the behavior of

r

x=2

. (a) Evaluate

\displaystyle \lim_{x\to 2^-} r(x)

and explain your reasoning based on the description of the curve for

x<2

. (b) Evaluate

\displaystyle \lim_{x\to 2^+} r(x)

and explain your reasoning, including an analysis of the boundedness of

\sin\bigl(\pi(x-2)\bigr)

x

approaches

2

. (c) Determine whether

\displaystyle \lim_{x\to 2} r(x)

exists. Justify your conclusion by synthesizing the results of parts (a) and (b). In your justification, discuss the types of discontinuity (if any) present at

x=2

. (d) Suppose

r

were re-defined at

x=2

by setting

r(2)=0

. Would this re-definition create a continuous function at

x=2

? Explain why or why not.

Related theory

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