Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) \(x \rightarrow \frac{\pi^{+}}{2}\left(\text { as } x \text { approaches } \frac{\pi}{2} \text { from the right }\right)\) (b) \(x \rightarrow \frac{\pi^{-}}{2}\left(\text { as } x \text { approaches } \frac{\pi}{2} \text { from the left }\right)\) (c) \(x \rightarrow-\frac{\pi^{+}}{2}\left(\text { as } x \text { approaches }-\frac{\pi}{2} \text { from the right }\right)\) (d) \(x \rightarrow-\frac{\pi^{-}}{2}\left(\text { as } x \text { approaches }-\frac{\pi}{2} \text { from the left }\right)\) \(f(x)=\tan x\)

Step-by-Step Solution

Answer

As \(x\) approaches \(\pi/2\) from the right, \(\tan x\) goes to negative infinity. As \(x\) approaches \(\pi/2\) from the left, \(\tan x\) goes to positive infinity. As \(x\) approaches \(-\pi/2\) from the right, \(\tan x\) goes to positive infinity. As \(x\) approaches \(-\pi/2\) from the left, \(\tan x\) goes to negative infinity.

Step 1: Graphing the function

First, use a graphing utility to graph the function \(f(x)=\tan x\). A graph of the function will show its general behavior for each interval from \(-\pi/2\) to \(\pi/2\).

Step 2: Observing the behavior as \(x\) approaches \(\pi/2\) from the right

Notice on the graph that as \(x\) approaches \(\pi/2\) from the right, the function \(f(x) = \tan x\) tends to negative infinity.

Step 3: Observing the behavior as \(x\) approaches \(\pi/2\) from the left

Note on the graph that as \(x\) approaches \(\pi/2\) from the left, the function \(f(x) = \tan x\) tends to positive infinity.

Step 4: Observing the behavior as \(x\) approaches \(-\pi/2\) from the right

Observe on the graph that as \(x\) approaches \(-\pi/2\) from the right, the function \(f(x) = \tan x\) tends to positive infinity.

Step 5: Observing the behavior as \(x\) approaches \(-\pi/2\) from the left

From the graph, as \(x\) approaches \(-\pi/2\) from the left, the function \(f(x) = \tan x\) tends to negative infinity.

Key Concepts

Behavior of the Tangent Function

The tangent function, represented as \( \tan x \), is one of the primary trigonometric functions. Unlike sine and cosine, the tangent function is periodic with period \( \pi \) and features vertical asymptotes.
These vertical asymptotes occur at integer multiples of \( \pi/2 \) where the value of tangent is undefined. Near these points, the tangent function shows dramatic behavior. As it approaches these vertical lines, the output of the function increases or decreases without bound. Consequently, we observe either positive or negative infinity, depending on the direction of approach.
The general behavior of the tangent can be summarized as follows:

• As \( x \) approaches \( \pi/2 \) from the right, \( \tan x \rightarrow -\infty \).
• As \( x \) approaches \( \pi/2 \) from the left, \( \tan x \rightarrow \infty \).
• As \( x \) approaches \( -\pi/2 \) from the right, \( \tan x \rightarrow \infty \).
• As \( x \) approaches \( -\pi/2 \) from the left, \( \tan x \rightarrow -\infty \).

This behavior illustrates the function's ability to oscillate between positive and negative directions rapidly, making it crucial to understand these tendencies for problem-solving.

Limits of Trigonometric Functions

Trigonometric limits help us predict the output of a function as the input approaches a particular value. With trigonometric functions, especially ones like the tangent with asymptotic behavior, these limits can differ based on the direction from which the function is approached.
For tangent, specifically, when evaluated around its asymptotes (like \( \pi/2 \) and \( -\pi/2 \)), the limits do not exist at the point because they diverge to infinity.
Consider how limits function for \( \tan x \) as it approaches its vertical asymptotes.

• If \( x \rightarrow \frac{\pi}{2}^{+} \), \( \tan x \rightarrow -\infty \).
• If \( x \rightarrow \frac{\pi}{2}^{-} \), \( \tan x \rightarrow \infty \).
• If \( x \rightarrow -\frac{\pi}{2}^{+} \), \( \tan x \rightarrow \infty \).
• If \( x \rightarrow -\frac{\pi}{2}^{-} \), \( \tan x \rightarrow -\infty \).

These behaviors showcase the special considerations needed for limits in trigonometric functions, underscoring the concept of approaching a point from different sides and how it impacts the function's behavior.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a particular point or line, most commonly when it approaches infinity or a vertical asymptote. For trigonometric functions, especially tangent, understanding asymptotes is key to predicting function behavior around critical points.
The tangent function has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. These asymptotes create boundaries where the function rapidly increases or decreases its values.

• Approaching from the right towards \( \pi/2 \), the function tends toward \(-\infty \).
• Approaching from the left, it tends toward \(\infty\).

The opposite behavior holds near \(-\pi/2\). Asymptotic behavior is critical in calculus, setting guidelines for where and how calculus tools apply. Knowing at which points a function will diverge and whether it does so to positive or negative infinity allows us to make accurate predictions regarding function values outside typical boundaries.
In essence, asymptotic analysis in the context of the tangent function provides vital insights into its infinite nature and practical use in mathematical calculations.