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

-1

, but

p(-1)

is not given. The table shows selected, independently rounded values.

$x$	$-1.4$	$-1.2$	$-1.05$	$-1.01$	$-0.99$	$-0.95$	$-0.8$
$p(x)$	$7.36$	$5.28$	$4.21$	$4.04$	$3.96$	$3.79$	$3.12$

$x$	$-1.4$	$-1.2$	$-1.05$	$-1.01$
Absolute rounding error	$\le 0.02$	$\le 0.02$	$\le 0.01$	$\le 0.005$

$x$	$-0.99$	$-0.95$	$-0.8$
Absolute rounding error	$\le 0.005$	$\le 0.01$	$\le 0.02$

Use the tables to investigate the behavior of

p

x=-1

. (a) Estimate

\displaystyle \lim_{x\to -1^-} p(x)

correct to three decimal places. Show numerically how you account for the stated rounding errors in forming your estimate. (b) Estimate

\displaystyle \lim_{x\to -1^+} p(x)

correct to three decimal places, again addressing the effect of rounding error. (c) Does

\displaystyle \lim_{x\to -1} p(x)

exist? Justify your answer with explicit reference to the estimates obtained in parts (a) and (b). (d) Suppose it is later discovered that the value at

x=-0.95

was mistakenly reported too low by

0.04

. Re-evaluate your conclusion in part (c), explaining whether it changes and why.

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