Question

a. Create a table of values of \(\tan (3 / x)\) for \(x=12 / \pi, 12 /(3 \pi), 12 /(5 \pi) \ldots . .12 /(11 \pi) .\) Describe the general pattern in the values you observe. b. Use a graphing utility to graph \(y=\tan (3 / x) .\) Why do graphing utilities have difficulty plotting the graph near \(x=0 ?\) c. What do you conclude about \(\lim _{x \rightarrow 0} \tan (3 / x) ?\)

Short answer

b. Describe any difficulties encountered when plotting the graph of \(y=\tan(3/x)\) near \(x=0\). c. What is the limit of \(\tan(3/x)\) as \(x\) approaches \(0\), and what conclusion can be drawn from this?

Step-by-step solution

Create the Table of Values

Calculate the values for \(\tan(3/x)\) at the required x-values: \(x= 12/\pi, 12/(3\pi), 12/(5\pi), 12/(7\pi), 12/(9\pi), 12/(11\pi)\).

Identify the Pattern

Observe the obtained values and describe the general pattern in those values.

Graph the Function

Use a graphing utility to plot the function \(y=\tan(3/x)\). Observe the graph and its behavior around \(x=0\).

Explain Difficulties in Plotting near \(x=0\)

Analyze the behavior of \(\tan(3/x)\) as \(x\) approaches \(0\). Discuss possible reasons graphing utilities might struggle to plot the graph of this function near \(x=0\).

Evaluate the Limit

Calculate the limit of \(\tan(3/x)\) as \(x\) approaches \(0\). If it does not exist, make a conclusion about this limit.

Key concepts

Tangent Function

The tangent function, denoted as \(\tan(\theta)\), is a fundamental trigonometric function. Understanding its behavior is crucial in analyzing the expression \(\tan(3/x)\). \(\tan(\theta)\) is defined as the ratio of the sine to the cosine function:

• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

This function exhibits a periodic nature, repeating its values over a regular interval, specifically every \(\pi\) radians.

As a result, when evaluating expressions like \(\tan(3/x)\), the periodicity comes into play. For different values of \(x\), the corresponding ratios might fluctuate significantly, particularly as \(x\) changes. This behavior is especially pronounced when \(x\) approaches values where \(\cos(3/x)\) approaches zero, leading to vertical asymptotes in the graph. Understanding the tangent function's fundamental properties helps in sketching its graph and predicting its behavior under certain transformations.

Graphing Utilities

Graphing utilities, such as graphing calculators or software, are vital tools for visualizing mathematical functions like \(y = \tan(3/x)\). These tools help students and mathematicians to understand complex relationships by providing a visual representation. However, these tools can struggle with certain functions near points of discontinuity or at sharp changes, like those seen in the tangent function.

When plotting \(y = \tan(3/x)\), graphing utilities might display difficulty near \(x = 0\). As \(x\) approaches zero, \(3/x\) increases without bound, causing \(\tan(3/x)\) to oscillate wildly because of its periodic nature. This can result in undefined values at points where \(x\) would cause the cosine component to be zero.

• Rapid oscillations lead to gaps in the plotted graph.
• Graphing tools may display vertical lines, signifying undefined values.

Understanding how these utilities work with such sections helps us utilize them appropriately, by adjusting viewing windows or considering theoretical results when visual depictions are inadequate.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches certain limits or infinity. With \(y = \tan(3/x)\), understanding its asymptotic behavior is key to interpreting the results of the graph and limit at \(x \to 0\).

As \(x\) gets closer to zero, \(3/x\) becomes extremely large, leading to rapid oscillations of the tangent function due to its periodicity. During these oscillations, each point where \(3/x = (2n+1)\frac{\pi}{2}\) (where \(n\) is an integer) results in an undefined \(\tan(\theta)\). This process creates vertical asymptotes.

Analyzing the limit \(\lim_{x \to 0} \tan(3/x)\) reveals that the function does not approach a specific value as \(x\) nears zero, due to these repeated vertical asymptotes and oscillations, implying that the limit does not exist.

• Tangent function's behavior near vertical asymptotes leads to infinite, repeating spikes.
• Understanding asymptotic limits aid in predicting and interpreting behavior of such complex functions.