For a positive integer

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

Problem

m

, define

s_m(x)=\begin{cases} \dfrac{x^{m}-1}{x-1}, & x\ne 1, \\ C, & x=1, \end{cases}

where

C

is a constant to be specified. A computer algebra system produced the following numerical data for

s_m

when

x

is near

1

$x$	$0.90$	$0.99$	$1.01$	$1.10$
$s_m(x)$	$2.710$	$2.9701$	$3.0301$	$3.331$

(a) Use the table to estimate

\displaystyle\lim_{x\to1}s_m(x)

. State the result to the nearest

0.001

. (b) Deduce the integer value of

m

that is consistent with the observed data. Explain your reasoning. (c) For that value of

m

, verify analytically that

\displaystyle\lim_{x\to1}s_m(x)

equals the value you estimated in part (a). (d) Determine the value of

C

required to make

s_m

continuous at

x=1

, and briefly justify your choice.