Consider a function whose graph consists of two straight-line segments as described below

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

Problem

Consider a function

g

whose graph consists of two straight-line segments as described below. • For

x<3

, the graph of

g

lies on the line through the points

(1,6)

and

(3,4)

. The point

(3,4)

is shown as an open circle, indicating it is not included on the graph. • For

x>3

, the graph of

g

lies on the line through the points

(3,1)

and

(5,3)

. The point

(3,1)

is solid, indicating it is included on the graph. There are no other portions of the graph. Answer the following questions about

g

. (a) Determine

\displaystyle \lim_{x\to 3^-}g(x)

. Show the computations that lead to your answer. (b) Determine

\displaystyle \lim_{x\to 3^+}g(x)

. Show the computations that lead to your answer. (c) Does

\displaystyle \lim_{x\to 3}g(x)

exist? Explain your reasoning, referring to the results from parts (a) and (b). (d) Evaluate

g(3)

and compare this value with the limit behavior you found in part (c). Describe the significance of any difference.