For each positive integer

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

Problem

a

, a function

P_a

is defined for all real numbers. Tables of

P_a(x)

for three different choices of

a

are provided. Table 1:

a=1

$x$	$20$	$40$	$80$	$160$
$P_1(x)$	$0.55$	$0.525$	$0.512$	$0.506$

Table 2:

a=2

$x$	$20$	$40$	$80$	$160$
$P_2(x)$	$1.10$	$1.05$	$1.025$	$1.013$

Table 3:

a=3

$x$	$20$	$40$	$80$	$160$
$P_3(x)$	$1.65$	$1.575$	$1.538$	$1.519$

Use the tables to analyze how the end behavior of

P_a

depends on the parameter

a

. (a) Using Table 1, estimate

\displaystyle \lim_{x\to\infty} P_1(x)

. Support your estimate numerically. (b) Repeat part (a) for

a=2

using Table 2. (c) Repeat part (a) for

a=3

using Table 3. (d) Based on the patterns you observed in parts (a)–(c), conjecture a general formula for

L(a)=\displaystyle \lim_{x\to\infty} P_a(x)

in terms of

a

. Provide a clear justification that references the numerical evidence from all three tables.