Consider the graph of a function in a neighborhood of described as follows

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

Problem

Consider the graph of a function

q

in a neighborhood of

x=-1

described as follows. • For

x<-1

, the graph follows the cubic

y=(x+1)^3+2

. • At

x=-1

, the point

(-1,2)

is removed from the graph (open circle), and instead a solid point is plotted at

(-1,5)

. • For

x>-1

, the graph follows the line

y=4x+9

. Assume no additional features occur near

x=-1

. (a) Estimate

\displaystyle \lim_{x\to -1^-} q(x)

and justify your estimate based on the left-hand description. (b) Estimate

\displaystyle \lim_{x\to -1^+} q(x)

and justify your estimate based on the right-hand description. (c) Does

\displaystyle \lim_{x\to -1} q(x)

exist? Explain fully, citing results from parts (a) and (b). (d) State

q(-1)

and discuss how the value of the function at

x=-1

compares to the two-sided limit.