Mathematics

Problem

A function

q

is defined for all real numbers. Its values for several large positive and negative inputs are shown below.

$x$	$10$	$20$	$30$	$40$	$50$
$q(x)$	$0.091$	$0.041$	$0.018$	$0.0082$	$0.0037$
$x$	$-10$	$-20$	$-30$	$-40$	$-50$
-----	-------	-------	-------	-------	-------
$q(x)$	$112$	$24{,}500$	$5.3\times10^6$	$1.1\times10^9$	$2.4\times10^{11}$

Answer the following questions about the limits of

q

. (a) Using the data for positive

x

, estimate

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

. Support your answer with numerical evidence. (b) Using the data for negative

x

, determine whether

\displaystyle \lim_{x\to-\infty} q(x)

appears to exist. Explain your reasoning clearly and cite specific values from the table. (c) Based on your results from parts (a) and (b), describe the qualitative end behavior of the graph of

q

x

approaches both

\infty

and

-\infty

. Include any horizontal asymptotes in your description.

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