Consider a function whose graph is described as follows

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

Problem

Consider a function

h

whose graph is described as follows. • For

x<1

, the graph coincides with the parabola

y=(x-1)^2-2

and includes an open circle at

(1,-2)

. • For

1\le x<3

, the graph is the constant segment

y=0

. • At

x=3

, there is a filled dot at

(3,2)

. • For

x>3

, the graph follows the curve

y=2+\dfrac{3}{x-3}

. Parts: (a) Evaluate

\displaystyle \lim_{x\to1^-}h(x)

\displaystyle \lim_{x\to1^+}h(x)

, and

h(1)

. (b) Evaluate

\displaystyle \lim_{x\to3^-}h(x)

\displaystyle \lim_{x\to3^+}h(x)

, and

h(3)

. (c) Using your results from parts (a) and (b), classify each discontinuity at

x=1

and

x=3

(removable, jump, or infinite), and justify each classification. (d) State all intervals on which

h

is continuous.