College Algebra, Chapter 4, 4.6, Section 4.6, Problem 10

If $\displaystyle r(x) = \frac{3x^2 + 1}{(x - 2)^2}$, then (a) Complete each table for the function. (b) Describe the behavior of the function near its vertical asymptote, based on Tables 1 and 2. (c) Determine the horizontal asymptote, based on Tables 3 and 4.

Table 1

$\begin{array}{|c|c|}
\hline\\
x & r(x) \\
\hline\\
1.5 & \\
\hline\\
1.9 & \\
\hline\\
1.99 & \\
\hline\\
1.999 & \\
\hline
\end{array} $

Table 2

$\begin{array}{|c|c|}
\hline\\
x & r(x) \\
\hline\\
2.5 & \\
\hline\\
2.1 & \\
\hline\\
2.01 & \\
\hline\\
2.001 & \\
\hline
\end{array} $

Table 3

$\begin{array}{|c|c|}
\hline\\
x & r(x) \\
\hline\\
10 & \\
\hline\\
50 & \\
\hline\\
100 & \\
\hline\\
1000 & \\
\hline
\end{array} $

Table 4

$\begin{array}{|c|c|}
\hline\\
x & r(x) \\
\hline\\
-10 & \\
\hline\\
-50 & \\
\hline\\
-100 & \\
\hline\\
-1000 & \\
\hline
\end{array} $


a.)

$\begin{array}{|c|c|c|c|}
\hline\\
\text{Table 1} & & \text{Table 2} & \\
\hline\\
x & r(x) & x & r(x) \\
\hline\\
1.5 & 31 & 2.5 & 79 \\
\hline\\
1.9 & 1183 & 2.1 & 1423 \\
\hline\\
1.99 & 128803 & 2.01 & 131203 \\
\hline\\
1.999 & 12988003 & 2.001 & 13012003\\
\hline
\end{array} $

b.) Based from the values obtained in the table, $r(x)$ approaches a
very big number of $x$ approaches 2 from either left or right side. It
means that $r(x)$ has a vertical asymptote at $x = 2$.

c.)

$\begin{array}{|c|c|c|c|}
\hline\\
\text{Table 3} & & \text{Table 4} & \\
\hline\\
x & r(x) & x & r(x) \\
\hline\\
10 & 4.7031 & -10 & 2.0903 \\
\hline\\
50 & 3.2556 & -50 & 2.7740 \\
\hline\\
100 & 3.1238 & -100 & 2.8836 \\
\hline\\
1000 & 3.0120 & -1000 & 2.9880 \\
\hline
\end{array} $

It shows from the table that $r(x)$ approaches 3 from either left or
right side as $x$ approaches a very big number. It means that $r(x)$
has a horizontal asymptote at $y = 3$.