A function is defined for values of near

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

Problem

A function

p

is defined for values of

x

near

x=3

. Selected values of

p

are shown in the table below.

$x$	$2.8$	$2.9$	$2.95$	$3.00$	$3.05$	$3.1$	$3.2$
$p(x)$	$1.496$	$1.663$	$1.741$	$1.822$	$1.903$	$1.978$	$2.130$

Let

h(x)=e^{x}

, and define the composite function

F(x)=h(p(x))

. Answer the following questions about the behavior of

F

x=3

. (1) Estimate

\displaystyle\lim_{x\to3}p(x)

using the data in the table, and state your estimate to three decimal places. (2) Using your answer from part (a), evaluate

\displaystyle\lim_{x\to3}F(x)

. Show your reasoning clearly. (3) Given that

p(3)=1.822

, compute

F(3)

and determine whether

F

is continuous at

x=3

. Justify your conclusion by referring to the formal definition of continuity at a point.