Consider the function defined by and the closed interval .

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

Problem

Consider the function

h

defined by

h(x)=\dfrac{(x-1)(x+4)}{x-3}

and the closed interval

[0,6]

. (a) Determine the exact

x

-coordinate(s) in

[0,6]

at which

h

is not continuous. Provide analytic justification using limits. (b) The graph of

h

is sketched on

[0,6]

with all discontinuities clearly marked. Describe the graphical behavior of

h

near each discontinuity you listed in part (a). (c) Explain whether

h

satisfies the hypotheses of the Intermediate Value Theorem on the interval

[0,6]

. State your reasoning in terms of continuity on the interval and any theorem conditions that fail.