Question

In Exercises 1 and \(2,\) complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$ \lim _{x \rightarrow \infty} \frac{6 x}{\sqrt{3 x^{2}-2 x}} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 10 & 10^{2} & 10^{3} & 10^{4} & 10^{5} \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} $$

Short answer

To obtain the short answer, compute the limit by observing the behavior of the function as \(x\) approaches infinity. The limit appears to converge to \(2\sqrt{3}\) as \(x\) becomes larger.

Step-by-step solution

Fill in the function values in the table

Given the function \(f(x) = \frac{6x}{\sqrt{3x^{2} - 2x}}\), we need to find the value of the function for each value of \(x\) in the given table. For \(x = 1, 10, 10^{2}, 10^{3}, 10^{4}, 10^{5}\), plug these into the function and calculate the corresponding function value.

Observe the behavior of the function as \(x\) approaches infinity

As can be seen from the table, the function value tends to steady as \(x\) becomes larger. This implies that the function approaches a specific value as \(x\) tends towards infinity.

Estimate the limit

Using the values in the table, the limit of the function as \(x\) approaches infinity can be estimated. If the function values seem to converge to a specific value as \(x\) becomes larger, that value will be the estimate for the limit.

Graph the function

To support your result, use a graphing utility to graph the function. From the graph, the behavior of the function as \(x\) approaches infinity could be observed.