Problem 26
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 \(\lim _{x \rightarrow 0} \tan (3 / x) ?\)
Step-by-Step Solution
Verified 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?
1Step 1: Create the Table of Values
Calculate the values for \(\tan(3/x)\) at the required x-values: \(x= 12/\pi, 12/(3\pi), 12/(5\pi), 12/(7\pi), 12/(9\pi), 12/(11\pi)\).
2Step 2: Identify the Pattern
Observe the obtained values and describe the general pattern in those values.
3Step 3: Graph the Function
Use a graphing utility to plot the function \(y=\tan(3/x)\). Observe the graph and its behavior around \(x=0\).
4Step 4: 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 \(x=0\).
5Step 5: 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 FunctionGraphing UtilitiesAsymptotic Behavior
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:
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.
- \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
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.
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.
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.
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.
Other exercises in this chapter
Problem 25
Evaluate the following limits. \(\lim _{x \rightarrow 1}\left(2 x^{3}-3 x^{2}+4 x+5\right)\)
View solution Problem 25
For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point. $$f(x)=2 x
View solution Problem 26
Determining limits analytically Determine the following limits or state that they do not exist. $$\lim _{t \rightarrow 5} \frac{4 t^{2}-100}{t-5}$$
View solution Problem 26
Suppose \(\lim _{x \rightarrow a} f(x)=L\). Prove that \(\lim _{x \rightarrow a}[c f(x)]=c L,\) where \(c\) is a constant.
View solution