A rational function is given. \[ r(x)=\frac{x}{x-7} \] (a) Complete each table for the function. (Round your answers to three decimal places where applicable.) Table 1 \begin{tabular}{|c|c|} \hline $\boldsymbol{x}$ & $\boldsymbol{r}(\boldsymbol{x})$ \\ \hline 6.5 & \\ 6.9 & \\ 6.99 & \\ 6.999 & \\ \hline \end{tabular} Table 3 \begin{tabular}{|r|r|} \hline $\boldsymbol{x}$ & $\boldsymbol{r}(\boldsymbol{x})$ \\ \hline 10 & \\ 50 & \\ 100 & \\ 1000 & \\ \hline \end{tabular} Table 2 \begin{tabular}{|c|c|} \hline $\boldsymbol{x}$ & $\boldsymbol{r}(\boldsymbol{x})$ \\ \hline 7.5 & \\ 7.1 & \\ 7.01 & \\ 7.001 & \\ \hline \end{tabular} Table 4 \begin{tabular}{|c|c|} \hline $\boldsymbol{x}$ & $\boldsymbol{r}(\boldsymbol{x})$ \\ \hline-10 & \\ -50 & \\ -100 & \\ -1000 & \\ \hline \end{tabular} (b) Describe the behavior of the function near its vertical asymptote, based on Tables 1 and 2. \[ r(x) \rightarrow \] as $x \rightarrow 7^{-}$and $r(x) \rightarrow$ as $x \rightarrow 7^{+}$. (c) Determine the horizontal asymptote, based on Tables 3 and 4. $y=$
Solution
Step 1 ：Given the rational function \(r(x) = \frac{x}{x-7}\), we are asked to fill in the tables for the given x-values.
Step 2 ：First, we substitute the x-values from Table 1 into the function to get the corresponding y-values. For example, when \(x = 6.5\), \(r(x) = \frac{6.5}{6.5-7} = -13.0\). Similarly, we find the y-values for \(x = 6.9, 6.99, 6.999\) to be \(-69.0, -699.0, -6999.0\) respectively.
Step 3 ：Next, we substitute the x-values from Table 2 into the function to get the corresponding y-values. For example, when \(x = 7.5\), \(r(x) = \frac{7.5}{7.5-7} = 15.0\). Similarly, we find the y-values for \(x = 7.1, 7.01, 7.001\) to be \(71.0, 701.0, 7001.0\) respectively.
Step 4 ：Then, we substitute the x-values from Table 3 into the function to get the corresponding y-values. For example, when \(x = 10\), \(r(x) = \frac{10}{10-7} = 1.333\). Similarly, we find the y-values for \(x = 50, 100, 1000\) to be \(1.163, 1.075, 1.007\) respectively.
Step 5 ：Finally, we substitute the x-values from Table 4 into the function to get the corresponding y-values. For example, when \(x = -10\), \(r(x) = \frac{-10}{-10-7} = 0.588\). Similarly, we find the y-values for \(x = -50, -100, -1000\) to be \(0.877, 0.935, 0.993\) respectively.
Step 6 ：Based on Tables 1 and 2, as \(x\) approaches 7 from the left (\(x \rightarrow 7^-\)), \(r(x)\) decreases without bound (\(r(x) \rightarrow -\infty\)). As \(x\) approaches 7 from the right (\(x \rightarrow 7^+\)), \(r(x)\) increases without bound (\(r(x) \rightarrow +\infty\)). This indicates that there is a vertical asymptote at \(x = 7\).
Step 7 ：Based on Tables 3 and 4, as \(x\) increases or decreases without bound (\(x \rightarrow \pm\infty\)), \(r(x)\) approaches 1 (\(r(x) \rightarrow 1\)). This indicates that there is a horizontal asymptote at \(y = 1\).
Step 8 ：\(\boxed{\text{Final Answer:}}\) The completed tables are as follows: \[\begin{array}{|c|c|} \hline x & r(x) \\ \hline 6.5 & -13.0 \\ 6.9 & -69.0 \\ 6.99 & -699.0 \\ 6.999 & -6999.0 \\ \hline \end{array}\] \[\begin{array}{|c|c|} \hline x & r(x) \\ \hline 7.5 & 15.0 \\ 7.1 & 71.0 \\ 7.01 & 701.0 \\ 7.001 & 7001.0 \\ \hline \end{array}\] \[\begin{array}{|c|c|} \hline x & r(x) \\ \hline 10 & 1.333 \\ 50 & 1.163 \\ 100 & 1.075 \\ 1000 & 1.007 \\ \hline \end{array}\] \[\begin{array}{|c|c|} \hline x & r(x) \\ \hline -10 & 0.588 \\ -50 & 0.877 \\ -100 & 0.935 \\ -1000 & 0.993 \\ \hline \end{array}\] The function has a vertical asymptote at \(x = 7\) and a horizontal asymptote at \(y = 1\).
