College Algebra, Chapter 4, 4.6, Section 4.6, Problem 8

If $\displaystyle r(x) = \frac{4x + 1}{x - 2}$, 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\\
2.0 & \\
\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 & -14 & 2.5 & 22 \\
\hline\\
1.9 & -86 & 2.1 & 94 \\
\hline\\
1.99 & -896 & 2.01 & 904 \\
\hline\\
1.999 & -8996 & 2.001 & 9004\\
\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 & 5.1250 & -10 & 3.25 \\
\hline\\
50 & 4.1875 & -50 & 3.8269 \\
\hline\\
100 & 4.0918 & -100 & 3.9118 \\
\hline\\
1000 & 4.0090 & -1000 & 3.9910\\
\hline
\end{array} $

It shows from the table that $r(x)$ approaches 4 from either left or right side as $x$ approaches a very big number. It means that $r(x)$ has a horizontal asymptote at $y = 4$.

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