Question

Numerical and Graphical Analysis In Exercises 3-8, use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=4+\frac{3}{x^{2}+2} $$

Short answer

Upon filling in the table, plotting the function and estimating the limit both numerically and graphically, it's observed that as \(x\) approaches infinity, the function \(f(x) = 4 + \frac{3}{{x^2+2}}\) appears to approach the value of 4.

Step-by-step solution

Completing the table

Use a graphing utility to complete the functions values at each \(x\). The function is \(f(x)=4+\frac{3}{x^{2}+2}\). Fill in the empty cells under \(f(x)\) by substituting the corresponding \(x\) values into the function.

Plot the Function

Use the same graphing utility to plot the function \(f(x)=4+\frac{3}{x^{2}+2}\). Use the \(x\) and \(f(x)\) pairs from the table as guide points. This will generate a curve or line, which is the graphical representation of the function.

Estimate the Limit

To estimate the limit of the function as \(x\) approaches infinity, observe the value towards which the \(f(x)\) values in your table are aiming, and the direction in which your graph is heading as \(x\) increases. The limit is the value that the function approaches as \(x\) approaches infinity.

Confirm the Limit

Confirm that the graphical limit aligns with numerical limit obtained from the table. Both methods should produce equivalent results for the limit of the function as \(x\) approaches infinity.