Problem 40 - Calculus Early Transcendentals

Question

Graph the function \(y=\sec x \tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \sec x \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \sec x \tan x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \sec x \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \sec x \tan x\)

Step-by-Step Solution

Answer

Answer: The limits of the function as x approaches \(\pm \pi/2\) are as follows: a. \(\lim_{x \rightarrow \pi / 2^{+}} \sec x \tan x=+\infty\). b. \(\lim_{x \rightarrow \pi / 2^{-}} \sec x \tan x=-\infty\). c. \(\lim_{x \rightarrow -\pi / 2^{+}} \sec x \tan x=-\infty\). d. \(\lim_{x \rightarrow -\pi / 2^{-}} \sec x \tan x=+\infty\).

Step 1: Identify the Domain of the Function

The domain of our function is the set of x-values for which the function is defined. The secant function is not defined for \(x = \frac{\pi}{2} + k \pi\), where k is any integer, because the cosine function takes a value of 0 for these points. This means that the secant function has vertical asymptotes at these points. The same applies to the tangent function. Therefore, the domain of our function is \(x \neq \frac{\pi}{2} + k \pi\).

Step 2: Graph the Function

We can graph the function \(y = \sec x \tan x\) by plotting points and identifying its behavior near the asymptotes. 1. Notice how the secant function has asymptotes at \(x = \frac{\pi}{2} + k \pi\), and the tangent function has vertical asymptotes at the same points. Since the secant and tangent functions get close to \(\infty\) or \(-\infty\) as they approach these asymptotes, so does the product of the two functions. 2. Everywhere else within the specified window, the product \(\sec x \tan x\) will be the product of the cosine and sine functions' reciprocals. Using graph plotting software like Desmos or a graphing calculator, plot the function within the given window. The graph should show the function intersecting the x-axis at x = 0 and the vertical asymptotes occurring at \(\pm \frac{\pi}{2}\).

Step 3: Analyze the Limits

Now that we have graphed the function, we can analyze the behavior of the function as x approaches different values: a. \(\lim_{x \rightarrow \pi / 2^{+}} \sec x \tan x\) From the graph, as x approaches \(\pi/2\) from the right, the function tends towards positive infinity. Therefore, \(\lim_{x \rightarrow \pi / 2^{+}} \sec x \tan x=+\infty\). b. \(\lim_{x \rightarrow \pi / 2^{-}} \sec x \tan x\) From the graph, as x approaches \(\pi/2\) from the left, the function tends towards negative infinity. Therefore, \(\lim_{x \rightarrow \pi / 2^{-}} \sec x \tan x=-\infty\). c. \(\lim_{x \rightarrow -\pi / 2^{+}} \sec x \tan x\) From the graph, as x approaches \(-\pi/2\) from the right, the function tends towards negative infinity. Therefore, \(\lim_{x \rightarrow -\pi / 2^{+}} \sec x \tan x=-\infty\). d. \(\lim_{x \rightarrow -\pi / 2^{-}} \sec x \tan x\) From the graph, as x approaches \(-\pi/2\) from the left, the function tends towards positive infinity. Therefore, \(\lim_{x \rightarrow -\pi / 2^{-}} \sec x \tan x=+\infty\).

Key Concepts

Asymptotes in Trigonometric Functions

Understanding asymptotes is essential when studying the behavior of trigonometric functions. Asymptotes are lines that the graph of a function approaches but never touches. Trigonometric functions can have vertical or horizontal asymptotes. For functions involving the secant (sec) and tangent (tan), vertical asymptotes are particularly common.

For example, in the function \(y = \text{sec} x \times \text{tan} x\), the sec and tan functions each have non-overlapping domains because they are undefined where their reciprocals, cosine (cos) and sine (sin), are zero. The points where cos and sin are zero—such as \(x = \frac{\pi}{2} + k\text{pi}\) for integer values of k—result in vertical asymptotes for sec and tan. These are the x-values where the function 'shoots' off to ±infinity and they are critical to understand the overall behavior of trigonometric functions.

Graphing Trigonometric Functions

Graphing trigonometric functions can appear daunting, but recognizing patterns and behaviors simplifies the process. Secant and tangent functions have periodic natures, causing their graphs to exhibit repeated behavior at regular intervals. For the graphing of \(y = \text{sec} x \times \text{tan} x\), we need to be cautious around its vertical asymptotes.

When graphing, it's essential to plot key points, like the intercepts and peak values, while also noting the function's continuity. It is also vital to sketch the asymptotes, as they help in visualizing where the function's graph cannot exist. These graphical representations aid in understanding the limits and predicting the function's behavior in different intervals.

Limit Behavior Analysis

Limits form the bedrock of calculus and help us analyze a function's behavior as it approaches a certain point. The limit behavior analysis of trigonometric functions, like \(y = \text{sec} x \times \text{tan} x\), involves determining what values the function approaches as x gets close to an asymptote or other critical points.

To analyze limits, we can look at the function's approach from both the left (\(x\) approaches the value from smaller numbers) and the right (\(x\) approaches from larger numbers). As seen in our example, the limits as x approaches \(\frac{\pi}{2}\) are different from the left and right, illustrating that these functions have distinct behaviors at those positions. Understanding limits is crucial as it also drives the concept of continuity and differentiability for more advanced calculus discussions.

Domain of Trigonometric Functions

The domain of trigonometric functions pertains to the set of all possible input values (usually x-values) for which the function is defined. In the case of the secant and tangent functions, their domains are restricted wherever their respective reciprocals, cosine and sine, equal zero because you cannot divide by zero.

In practical terms, this means that functions like \(y = \text{sec} x \times \text{tan} x\) will not have real values at certain x-values—specifically where \(x = \frac{\pi}{2} + k\text{pi}\) for any integer value of k. It is important to identify the domain before proceeding with any graphing or limit analysis, as it influences the range of x-values you consider for your calculations and graphical representations.