Question

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

Short answer

Refer to the graph created by the graphing utility to identify and determine the left-sided limit as \(x\) approaches 1. This value is the answer to the exercise.

Step-by-step solution

Graph the function

Input the function \( f(x)=\frac{x^{3}-1}{x^{2}+x+1} \) into the graphing utility. Ensure it is accurately plotted.

Identify the limit from the left

Look at the graph where \(x\) is slightly less than 1 (hence the symbol \(x \rightarrow 1^{-}\)). This is the left-sided limit.

Determine the value of the one-sided limit

Consider the values of the function as \(x\) approaches 1 from the left. The graph tends to a certain value - that value is the left-hand limit of the function \(f(x)\) at \(x = 1\).

Key concepts

Graphing Utility

Understanding the behavior of functions graphically is a crucial aspect in mathematics, especially when dealing with complex functions. A graphing utility, such as a graphing calculator or software, allows students to visualize the function by plotting its graph. This is particularly handy when identifying limits, as it offers an intuitive look at how the function behaves as it approaches a specific value. To use a graphing utility effectively, one must input the function correctly and adjust the viewing window to capture the relevant part of the graph.

For example, graphing the function \(f(x)=\frac{x^{3}-1}{x^{2}+x+1}\) provides a visual representation that can help determine the one-sided limit as \(x\) approaches 1 from the left. By zooming in on the graph around \(x=1\), the trend of the function's value can be observed, allowing students to make an educated guess about the limit before proceeding to more formal methods.

Limits in Calculus

In calculus, the concept of limits helps us understand the behavior of functions as they approach a certain point. The limit of a function at a particular point is the value the function output (or \(y\)) tends toward as the input (or \(x\)) approaches that point. One-sided limits are particular in that they only consider the function's behavior from one side—either from the left (\(x \rightarrow c^{-}\)) or the right (\(x \rightarrow c^{+}\)).

The exercise involves finding a one-sided limit from the left. By examining the graph near \(x=1\) and observing the y-values as \(x\) approaches 1 from the left, we can determine the one-sided limit, denoted as \(\lim _{x \rightarrow 1^{-}} f(x)\). The visualization on a graph is a preliminary step, which must be followed by analytical or numerical evaluation for precise results.

Rational Functions

A rational function is a fraction of two polynomials. In the given exercise, the function \(f(x)=\frac{x^{3}-1}{x^{2}+x+1}\) is a rational function because both the numerator \(x^{3}-1\) and the denominator \(x^{2}+x+1\) are polynomials. It's important to note that rational functions can behave unpredictably near points where the denominator approaches zero, leading to discontinuities or vertical asymptotes.

To analyze the behavior of a rational function at these points, one-sided limits are particularly useful. They can indicate whether the function is approaching infinity or a finite value as \(x\) approaches a particular point. When using graphing utilities, always pay special attention to the areas where the denominator is almost zero, as these regions often require a more thorough investigation to understand the function's limit.