Consider a function whose values near are given in the table

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Consider a function

p

whose values near

x=0

are given in the table.

$x$	$-0.1$	$-0.01$	$-0.001$	$0.001$	$0.01$	$0.1$
$p(x)$	0.95	$-0.98$	0.999	$-1.002$	0.97	$-0.93$

(a) Based on the numerical evidence, decide whether

\displaystyle \lim_{x\to 0} p(x)

exists. Clearly articulate your conclusion and support it with data from the table. (b) Provide a rigorous argument, written in complete sentences, that explains why the observed oscillatory pattern in the values of

p(x)

prevents the limit from existing. (c) A new function

q

is defined by

q(x)=x^2 p(x)

. Without computing

q

analytically, a student generates the following supplementary table for

q

near

x=0

$x$	$-0.1$	$-0.01$	$-0.001$	$0.001$	$0.01$	$0.1$
$q(x)$	0.0095	$-0.000098$	0.000000999	$-0.000001002$	0.00000097	$-0.0093$

Use the data to estimate

\displaystyle \lim_{x\to 0} q(x)

and justify why this limit is different from the limit analyzed in part (a).

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