Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Tell whether each statement is true or false. If false, tell why. The least positive number \(k\) for which \(x=k\) is an asymptote for the tangent function is \(\frac{\pi}{2}\).

Step-by-Step Solution

Answer

True, the least positive \( k \) is \( \frac{\pi}{2} \).

Step 1: Understand the Tangent Function Asymptotes

The tangent function, \( \tan(x) \), has vertical asymptotes where it is undefined. These occur at \( x = \frac{\pi}{2} + n\pi \) for integer \( n \). This means the function goes to infinity as it approaches these points.

Step 2: Identify the Smallest Positive Asymptote

The equation \( x = \frac{\pi}{2} + n\pi \) defines the vertical asymptotes for the tangent function. The smallest positive solution for \( x \) is when \( n = 0 \), giving \( x = \frac{\pi}{2} \).

Step 3: Evaluate the Statement

The statement claims the least positive \( k \) for which \( x=k \) is an asymptote of the tangent function is \( \frac{\pi}{2} \). Since our analysis shows \( \frac{\pi}{2} \) indeed is the smallest positive value for an asymptote, the statement is true.

Key Concepts

Understanding Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. In the context of functions, they typically refer to where the function becomes unbounded or undefined. These critical points often appear in rational functions and some trigonometric functions. The tangent function, represented by \( \tan(x) \), is one such example where asymptotes are present. The curves of the tangent function get indefinitely close to these vertical lines as \( x \) approaches certain values, but the graph never touches or crosses them. Understanding where these lines are located helps predict and graph the behavior of functions in mathematical studies.

Trigonometric Functions Overview

Trigonometric functions are fundamental in mathematics with roots in measuring angles and understanding circular motion. They are primarily defined on the unit circle and include sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent. These functions repeatedly oscillate in a wave-like manner and have unique properties that make them applicable in various fields, from physics to engineering.

For example, the sine and cosine functions are continuous and defined for all real numbers. However, the tangent function, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), becomes undefined wherever cosine is zero, as division by zero is not possible. This results in the formation of vertical asymptotes, where the function is undefined.

Understanding Vertical Asymptotes

Vertical asymptotes are specific types of asymptotes that occur at the values of \( x \) where a function is undefined or its values reach infinity. For the tangent function, these vertical asymptotes happen because the tangent function is based on the ratio of sine to cosine. When \( \cos(x) = 0 \), such as at \( x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \) and so on, the tangent function cannot compute a value.

The vertical asymptotes for the tangent function are mathematically described by the equation \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer. This means they repeat periodically after every \( \pi \) units along the \( x \)-axis. Understanding these vertical asymptotes is crucial, as they inform us where the function will blow up and greatly influence the graph's shape.