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. $$f(x)=\frac{\tan x}{\sin x}$$

Short answer

The vertical asymptotes of the function \(f(x) = \frac{\tan x}{\sin x}\) occur at \(x = \frac{π}{2} + nπ\), where n is an integer. As \(x\) approaches these asymptotes, \(f(x)\) tends to \(+∞\) or \(-∞\). For each asymptote, if \(n\) is even, \(f(x)\) tends to \(+∞\) on the left and \(-∞\) on the right. If \(n\) is odd, \(f(x)\) tends to \(-∞\) on the left and \(+∞\) on the right.

Step-by-step solution

Simplify the function

We start by simplifying the function. Notice that \(\tan x = \frac{\sin x}{\cos x}\). Substituting this into our function gives: \(f(x) = \frac{\frac{\sin x}{\cos x}}{\sin x} = \frac{1}{\cos x} = \sec x\). Hence, the original function simplifies to the secant function.

Determine the vertical asymptotes

Next, we need to find out where the secant function is not defined, those are the x-values for the vertical asymptotes. The secant function \(\sec x\) is undefined whenever \(\cos x = 0\). The cosine function equals zero at \(x = \frac{π}{2} + nπ\), where \(n\) is an integer. Therefore, the vertical asymptotes for \(f(x)\) are at \(x = \frac{π}{2} + nπ\).

Describe the behavior near asymptotes

Finally, we describe the behavior of \(f(x)\) near these asymptotes. As \(x\) approaches these asymptotes from the left or right, \(f(x)\) tends to \(+∞\) or \(-∞\). For each asymptote, if \(n\) is even, \(f(x)\) tends to \(+∞\) on the left and \(-∞\) on the right. If \(n\) is odd, \(f(x)\) tends to \(-∞\) on the left and \(+∞\) on the right. This is characteristic behavior of the secant function.

Key concepts

Secant Function

The secant function, denoted as \(\sec x\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function, so \(\sec x = \frac{1}{\cos x}\). The secant function can be derived directly from the unit circle, where the secant of an angle is the length of the segment drawn from the center of the circle to a point on the circle intersecting the terminal side in a way that it creates a horizontal line.

Understanding the graphical representation of the secant function helps to identify its characteristic behavior. The function creates a wave-like pattern with repeating intervals, exhibiting vertical asymptotes where the cosine function is zero. As we've seen in the exercise, this happens at \(x = \frac{\pi}{2} + n\pi\) where \(n\) is an integer. These points are crucial for understanding the secant's behavior, as they are where the function's values tend to infinity, which are the vertical asymptotes.

Trigonometric Functions

Trigonometric functions are a set of functions that relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cotangent (\(\cot\)), secant (\(\sec\)), and cosecant (\(\csc\)). The functions have various properties and graphs that display periodic behavior. This means they repeat their values in regular intervals, which are known as periods.

Understanding trigonometric functions is not only key in geometry and calculus but also in various practical applications such as physics, engineering, and waves. Their role in describing the motion around a circle is fundamental in creating models for periodic phenomena.

Limits and Continuity

Limits and continuity are foundational concepts in calculus. The limit of a function describes the behavior of the function as the input values approach a certain point. It is not about the value at the point but rather what value the function is approaching. We often use limits to define more complex concepts like derivatives and integrals.

Continuity, on the other hand, refers to the idea that a small change in the input of a function leads to a small change in the output. A function is considered continuous at a point if the limit as the input approaches the point equals the function's value at that point. This becomes highly relevant in our exercise where we examine the behavior of a function near the points where it is not continuous, that is, around the vertical asymptotes where limits tend to infinity.

Asymptotic Behavior

Asymptotic behavior in mathematics refers to the way functions behave near certain points or as the input values go towards infinity. It is particularly useful for describing the behavior of functions near points they are not defined, such as vertical asymptotes in our exercise's context. A vertical asymptote occurs at a point on the x-axis where a function’s value increases or decreases without bound.

The vertical asymptotes signal that the function approaches infinity or negative infinity; the function will never touch the asymptote but gets infinitely close. Recognizing this behavior helps predict the function's behavior and understand the nature of discontinuities, as seen with the secant function's tendency to approach positive or negative infinity as it nears points where the cosine function is zero.