A function is represented by the following verbal description of its graph in a…

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

Problem

A function

f

is represented by the following verbal description of its graph in a neighborhood of

x=2

. • For

x<2

, the graph is a straight line with slope

-1

passing through the point

(0,3)

. • At

x=2

, there is an open circle at the point

(2,1)

. • For

x>2

, the graph follows the parabola

y=(x-2)^2+1

and is closed at its initial point

(2,1)

. Assume no other unusual behavior occurs near

x=2

. (a) Estimate

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

and explain how the left–hand behavior of the graph supports your estimate. (b) Estimate

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

and explain how the right–hand behavior of the graph supports your estimate. (c) Determine whether

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

exists. Justify your conclusion using the results from parts (a) and (b).