A function is defined on the closed interval with the graph described below

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

Problem

A function

p

is defined on the closed interval

[-4,4]

with the graph described below. • For

-4\le x<-2

, the graph is the straight line joining

(-4,5)

(-2,1)

and includes the point at

x=-4

but not the point at

x=-2

. • At

x=-2

, there is a filled point at

(-2,0)

. • For

-2<x<1

, the graph is the curve

y=\sqrt{x+2}

. • At

x=1

, there is an open circle at

(1,3)

and a filled square at

(1,2)

. • For

1<x\le 4

, the graph is the straight line

y=-x+4

, including the endpoint at

x=4

but excluding

x=1

. Use this information to answer the following. (i) List all points in

[-4,4]

at which

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

does not exist, and justify each answer. (ii) List all points where the limit exists but

p

is not continuous. For each such point, state the common value of the one–sided limits and the value of

p(a)

. (iii) Suggest the smallest possible set of

x

–coordinates where redefining

p

(changing only the function value, not the graph elsewhere) would make

p

continuous on

[-4,4]

. Explain your choice.