Question

In Exercises \(27-34,\) (a) find the vertical asymptotes of the graph of \(f(x) .(\) b) Describe the behavior of \(f(x)\) to the left and right of each vertical asymptote. $$f(x)=\frac{\cot x}{\cos x}$$

Short answer

The vertical asymptotes of the function \(f(x) = \frac{\cot x}{\cos x}\) are \(x = n\pi\), where \(n\) is any integer. The function will approach \(-\infty\) as \(x\) approaches \(n\pi^-\) and will approach \(\infty\) as \(x\) approaches \(n\pi^+\).

Step-by-step solution

Convert the Function to a Standard Form

The function can be simplified by converting cotangent to a ratio of sine and cosine and then simplifying the expression, applying the identity \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\). Therefore, the given function \(f(x)=\frac{\cot x}{\cos x}\) can be converted to standard form \(f(x)= \frac{\cos x}{\sin x \cdot \cos x}\). Simplifying gives \(f(x)= \frac{1}{\sin x}\), or \(f(x) = \csc x\).

Find the Vertical Asymptotes

To find the vertical asymptotes, equate the denominator to zero. For the function \(f(x) = \csc x\), the denominator is \(\sin x\). Setting \(\sin x = 0\), we get \(x = n\pi\), where \(n\) is any integer. Thus, the vertical asymptotes are \(x = n\pi\), \(n\) is an integer.

Describe the Behavior to the Left and Right of Each Asymptote

Here, one needs to analyze the behavior on the immediate left and right of each asymptote. The left and right behavior of the function for each asymptote will be inverse due to the nature of the cosecant function. For \(x = n\pi\), as \(x\) approaches \(n\pi^-\), \(f(x)\) approaches \(-\infty\), and as \(x\) approaches \(n\pi^+\), \(f(x)\) approaches \(\infty\).