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. $$f(x)=\cot x$$

Short answer

The vertical asymptotes of the function \(f(x)=\cot x\) occur at \(x = n\pi\), where \(n\) is an integer. \(f(x)\) tends to positive infinity to the left of each vertical asymptote and tends to negative infinity to the right of each vertical asymptote.

Step-by-step solution

Finding Vertical Asymptotes

The cotangent function, \(f(x)=\cot x\), has vertical asymptotes wherever the function is undefined, that is, where the denominator of the cotangent function equals zero. The function \(\cot x\) is undefined where \(x = n\pi\), for all integers \(n\). Therefore, those are the vertical asymptotes.

Behavior to the Left of Each Vertical Asymptote

As \(x\) approaches \(n\pi\) from the left (i.e. as \(x\) gets close to \(n\pi\) but is just a tiny bit less), \(\cot\) x increases without bound. So, the behavior of the function \(f(x) = \cot x\) to the left of each vertical asymptote is that it tends to positive infinity.

Behavior to the Right of Each Vertical Asymptote

As \(x\) approaches \(n\pi\) from the right (i.e., as \(x\) gets close to \(n\pi\) but is just a tiny bit more), \(\cot x\) decreases without bound. So, the behavior of the function \(f(x) = \cot x\) to the right of each vertical asymptote is that it tends to negative infinity.

Key concepts

Cotangent Function

The cotangent function, represented as \(f(x) = \cot x\), is one of the six fundamental trigonometric functions. It's defined as the ratio of the adjacent side to the opposite side in a right-angled triangle or, in terms of the unit circle, as the reciprocal of the tangent function. Therefore, \(\cot x = \frac{1}{\tan x}\), or equivalently, \(\cot x = \frac{\cos x}{\sin x}\), where \(\cos x\) and \(\sin x\) are the cosine and sine functions respectively.

The cotangent curve is periodic, meaning it repeats its shape every \(\pi \text{ radians}\). This is because the sine and cosine functions that compose it are also periodic with a period of \(2\pi\). However, the cotangent function only takes half the period of sine and cosine to complete one cycle, thus its period is \(\pi\).

Understanding the behavior of the cotangent function leads to recognizing patterns such as its symmetries and periodicity, which are essential concepts for analyzing its graph and applications in various fields including physics and engineering.

Undefined Points in Trigonometric Functions

In trigonometry, undefined points, or points of discontinuity, occur where a function loses its ability to provide a definite value. For the cotangent function, these points arise specifically where the sine function, which serves as the denominator when \(\cot x = \frac{\cos x}{\sin x}\), equals zero. The sine function is zero at multiples of \(\pi\), hence the points \(x = n\pi\), where \(n\) is an integer, are where the cotangent function is undefined.

This is because dividing by zero is undefined in conventional mathematics. Each of these points results in a vertical asymptote on the graph of the cotangent function, marking places where the function approaches infinity. Knowing these points of discontinuity is vital for correctly sketching the graph of any trigonometric function and for understanding their behavior in real-world applications, like signal processing or wave dynamics.

Behavior Around Asymptotes

The behavior of a function around an asymptote can be analyzed as the function's value approaches infinity or negative infinity. Vertical asymptotes occur in the graph of \(f(x) = \cot x\) at \(x = n\pi\), and as the function nears these points, its behavior changes dramatically.

To the left of each vertical asymptote (when \(x\) is less than \(n\pi\)), the cotangent function increases rapidly as it heads towards positive infinity. Conversely, to the right of each vertical asymptote (when \(x\) is greater than \(n\pi\)), the cotangent function decreases rapidly towards negative infinity. This is indicative of an inversely proportional relationship, as one quantity increases, the other decreases.

Understanding this behavior is crucial for students who need to predict the value of trigonometric functions near undefined points. For instance, in physics, this information can be used to describe resonance phenomena where a small input at the right frequency can cause large oscillations due to the behavior around asymptotes.