Question

In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\frac{x^{2}+x+1}{x^{3}-1} \\ \lim _{x \rightarrow 1^{+}} f(x) \end{array} $$

Short answer

Without a specific graph, a precise numerical answer can't be provided. However, following these steps using a graphing utility will provide an accurate one-sided limit for \(x \rightarrow 1^{+}f(x)\) in the function \(f(x)=\frac{x^{2}+x+1}{x^{3}-1}\). Make sure to observe how the function behaves near \(x=1\) when determining your estimate.

Step-by-step solution

Graphing the function

Begin by inputting the function \(f(x)=\frac{x^{2}+x+1}{x^{3}-1}\) into a graphing utility. Take note of the graph's behavior, particularly near \(x=1\) as this will inform the evaluation of the one-sided limit.

Examining the graph

Examine the graph around \(x=1\). Observe if there is a discontinuity at \(x=1\) and how the function behaves as \(x\) approaches 1 from the right.

Calculating the one-sided limit

From the graph, estimate the value of the function as \(x\) approaches 1 from the right. Because we are dealing with a one-sided limit from the positive side, we are interested in what value the function appears to be nearing as \(x\) becomes closer and closer to 1 from values greater than 1.