Question

Numerical and Graphical Analysis In Exercises \(9-12\) , determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-3\) from the left and from the right by completing the table. Use a graphing utility to graph the function to confirm your answer. $$ \begin{array}{|l|l|l|}\hline x & {-3.5} & {-3.1} & {-3.01} & {-3.001} & {-3} \\\ \hline f(x) & {} & {} & {} & {} & {?} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|}\hline x & {-2.999} & {-2.99} & {-2.9} & {-2.5} \\\ \hline f(x) & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\cot \frac{\pi x}{3} $$

Short answer

Due to the behavior of the cotangent function, \(f(x) = \cot \frac{\pi x}{3}\) will approach \(\pm \infty\) as \(x\) approaches -3 from either side. This can be observed by computing the function values for the given x-values and graphing the function.

Step-by-step solution

Determine \(f(x)\) for \(x < -3\)

First we substitute \(x = -3.5, -3.1, -3.01,\) and \(-3.001\) into the function \(f(x)\) to see if the function approaches infinity or negative infinity as \(x\) gets closer to -3 from the left.

Determine \(f(x)\) for \(x > -3\)

Next, substitute \(x = -2.999, -2.99, -2.9, -2.5\) into the function \(f(x)\). This will show if the function is approaching infinity or negative infinity from the right as \(x\) tends to -3.

Graphical Analysis

Use a graphing calculator or online graphing tool to plot the function \(f(x) = \cot \frac{\pi x}{3}\). Verify if there is a vertical asymptote at \(x = -3\), which would suggest that the function does approach either infinity or negative infinity as \(x\) approaches -3 from either direction.

Key concepts

Numerical Analysis

Numerical analysis is a crucial approach when dealing with limits and continuity, especially when the exact behavior of a function is not immediately obvious. In our exercise, we want to understand how the function \( f(x) = \cot \frac{\pi x}{3} \) behaves as \( x \to -3 \) from both the left and the right.
To begin, numerical analysis involves substituting values of \( x \) approaching \(-3\) into the function and observing the resulting \( f(x) \) values. When \( x \) is less than \(-3\), such as \(-3.5, -3.1, -3.01,\) and \(-3.001\), by plugging these values into the function, we compute the tangent inverse.

• With each substitution, keep track of whether \( f(x) \) results in significantly large positive or negative numbers.
• Similarly, for \( x > -3 \), using values like \(-2.999, -2.99, -2.9,\) and \(-2.5\), observe how \( f(x) \) changes with these inputs.

By doing so, numerical analysis helps in determining whether the function approaches \( \infty \) or \( -\infty \) as \( x \to -3 \). This step-by-step approach offers a clear insight into the behavior of the function at crucial points.

Graphical Analysis

Graphical analysis is another powerful tool to understand functions and their behaviors. In this exercise, plotting the graph of \( f(x) = \cot \frac{\pi x}{3} \) can give us visual confirmation of the function's tendencies as \( x \to -3 \).
Using graphing utilities, such as a graphing calculator or software, we can plot the function across a range that includes \( x = -3 \). By viewing this graph, we can visually identify trends where the function might spike or dip suddenly, indicating potential approaches to infinity or negative infinity.
Key steps in graphical analysis include:

• Checking for the direction of spikes near \( x = -3 \), which can highlight if the function is unbounded in positive or negative directions.
• Noticing any repeating cycles or patterns which are characteristic of trigonometric functions like the cotangent.

Through graphical analysis, we confirm the presence of behaviors suggested by numerical analysis, providing a comprehensive picture of the function's continuity or lack thereof.

Vertical Asymptotes

Vertical asymptotes occur where a function approaches infinity or negative infinity as it gets arbitrarily close to a specific value. For our function \( f(x) = \cot \frac{\pi x}{3} \), a vertical asymptote can manifest where the cotangent function is undefined, particularly where the denominator of the cotangent's reciprocal sine function equals zero.
To identify vertical asymptotes, find where the sine denominator equates to zero:- Solve \( \sin\frac{\pi x}{3} = 0 \), which leads to specific points of indefiniteness. In this exercise, \( x = -3 \) is such a point.
Once verified through mathematical calculation, the vertical asymptotes:

• Serve as key indicators for where the function will shoot to infinity.
• Help in understanding the limits of contiguous regions on the graph.

Consequently, vertical asymptotes are pivotal in comprehending how \( f(x) \) behaves as \( x \to -3 \), indicating the function's discontinuity at that point.