A function is defined on the interval

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

Problem

A function

p

is defined on the interval

-2\le x\le4

. Its graph is described piece-wise as follows. • On

-2\le x<0

p

is the graph of the quadratic

y=(x+1)^2-1

. • At

x=0

, there is a filled point at

(0,2)

, but no other portion of the quadratic is present at that

x

-value. • On

0<x<2

p

is the graph of the line segment joining the points

(0,1)

(open point) and

(2,3)

(open point). • For

x=2

, there is a filled point at

(2,0)

. • For

2<x\le4

p

is the graph of a decreasing exponential curve that passes through

(2,3)

(open point) and

(4,1)

(closed point). Using this information, analyze the existence of two-sided limits of

p

over the specified domain. (a) List all values of

c

in the interval

-2\le c\le4

at which

\displaystyle \lim_{x\to c} p(x)

exists. Organize your answer in increasing order. (b) For each value of

c

you listed in part (a), state the corresponding limit value

L=\displaystyle \lim_{x\to c} p(x)

. (c) Provide a concise justification for one value of

c

at which the limit exists and one value at which the limit fails to exist, explicitly referencing the left-hand and right-hand behavior of

p

at those points.