Question

In Exercises 51–54, use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ \begin{array}{l}{f(x)=\frac{x-3}{x^{2}-9}} \\ {\lim _{x \rightarrow 3} f(x)}\end{array} $$

Short answer

The graph of the function \(f(x)=\frac{x-3}{x^{2}-9}\) has vertical asymptotes at \(x=-3\) and \(x=3\) indicating that these values are not included in the domain of the function. The limit as \(x \rightarrow 3\) of \(f(x)\) does not exist as the function approaches '\(-\infty\)' from both the left and the right. The domain of the function is all real numbers except \(x=-3,3\). Graphical representation of functions, though informative, may be misleading. Analytical examination provides clarity, precision, and comprehensive understanding of a function.

Step-by-step solution

Graph the Function

Using a graphing utility, graph the function \(f(x)=\frac{x-3}{x^{2}-9}\). The graph will help visualize the behavior of the function, especially around \(x=3\).

Estimate the Limit

By observing the behavior of the function as \(x\) approaches 3, estimate the limit. In this case, the function appears to approach '\(-\infty\)' from the left and '\(-\infty\)' from the right, hence the limit as \(x \rightarrow 3\) of \(f(x)\) does not exist because the function approaches different values from the left and right.

Determine the Domain

The domain of this function is all real numbers except for the values that make the denominator zero, this is when \(x^2-9=0\). Hence, \(x=-3,3\) are not in the domain. You can observe this by looking at the graph and noticing the vertical asymptotes at \(x=-3\) and \(x=3\).

Discuss the Importance of Analytical Examination

While a graphing utility provides a visual representation of the function and helps in identifying general trends, limitations, and discontinuities, it may not give a complete picture, particularly for complex functions or in marginal cases. Areas of discontinuity may not be accurately represented due to resolution limitations. Also, a graph fails to provide a precise value of the limit or clearly identify the domain. Therefore, examining a function analytically along with graphically is crucial. Analytical examination allows for precise calculations and helps overcome the resolution constraints of a graphing utility. It aids in a clear understanding of the function's behavior at different points, offering a more comprehensive picture.

Key concepts

Graphing Functions

Functions can often be quite complex, and looking at their graphs can provide significant insight into their behavior. When you graph a function like \(f(x) = \frac{x-3}{x^2-9}\), it's crucial to pay attention to areas where the function exhibits unusual behavior, such as sharp turns or jumps.
One essential aspect to observe is how the function behaves as it approaches certain points, such as where the denominator equals zero. Graphing utilities help us visualize these aspects by generating a plot of the function based on calculations at numerous points along the x-axis.
While graphs can showcase limits and asymptotic behavior, they should be used as a complement to analytical evaluations to avoid misinterpretations due to resolution or scaling issues.

• Use graphs for a visual understanding.
• Look for asymptotes or discontinuities.
• Analyze the plot closely near critical points.

Domain of Functions

The domain of a function is the set of all input values (x-values) for which the function is defined. For the function \(f(x) = \frac{x-3}{x^2-9}\), determining this domain requires identifying values which make the denominator zero.
Set the denominator equal to zero: \(x^2-9=0\). Solving this, \(x=3\) or \(x=-3\), these points cause the function to be undefined.
Thus, the domain of \(f(x)\) includes all real numbers except \(x = 3\) and \(x = -3\). It's important to note that just relying on a graph to determine the domain can lead to errors, as the graph may not highlight these undefined points effectively due to resolution or display limitations.

• Determine the domain by setting the denominator to zero.
• Exclude values where the function is undefined.
• Graphs can visually suggest domain issues but need analytical backing.

Analytical Examination

While graphing is a powerful tool, analytical examination complements it by providing a detailed, precise analysis of a function. For \(f(x) = \frac{x-3}{x^2-9}\), calculating values such as limits analytically helps in confirming what the graph suggests.
Analytic approaches involve algebraic manipulation, which can reveal aspects like removable discontinuities, hidden within the function's expression. This method also enables exact calculation of limits, confirming the behavior as \(x \rightarrow 3\) or other critical points.
Analysis enhances understanding by detailing each step and resolving ambiguities that a visual graph might create. Relying solely on a graph may mask minute but important details due to scale or pixelation, which analytical examination can clarify.

• Use algebraic methods to understand the function.
• Confirm suspected behavior indicated by graphs.
• Identify limits and discontinuities accurately.

Discontinuities in Functions

In calculus, understanding discontinuities is key to grasping a function's behavior. Discontinuities occur when there are breaks, jumps, or asymptotes in the graph of a function. For the function \(f(x) = \frac{x-3}{x^2-9}\), we encounter a discontinuity at \(x=3\) and \(x=-3\) because the function is undefined at these points.
Discontinuities can be of different types: removable, jump, or infinite. In this example, the points \(x=3\) and \(x=-3\) lead to vertical asymptotes, a type of infinite discontinuity, where the function's value heads towards infinity as you approach these x-values.
By examining both the analytic form and the graph, you can distinguish these discontinuities effectively. It's also important to note that errors in detecting discontinuities can stem from not analyzing the function closely or relying solely on a graph.

• Identify breaks in the graph as signs of discontinuities.
• Understand types: removable, jump, infinite.
• Use graphs and formulas for clearer insight.