Problem 28 - Calculus of a Single Variable

Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ g(\theta)=\frac{\tan \theta}{\theta} $$

Step-by-Step Solution

Answer

The vertical asymptotes of the function \( g(\theta) = \frac{\tan \theta}{\theta} \) are at \( \theta = 0 \) and \( \theta = n\pi, n\neq 0 \).

Step 1: Identify the function

The function to work with is \( g(\theta) = \frac{\tan \theta}{\theta} \). This function is undefined at \(\theta = 0\) and \(\theta = n\pi, n\neq 0\) where \( n \) is an integer.

Step 2: Find the vertical asymptotes

Vertical asymptotes occur at values of \( \theta \) where the function \( g(\theta) = \frac{\tan \theta}{\theta} \) is undefined. Here, the function is undefined when \( \theta = 0 \) and \( \theta = n\pi, n\neq 0 \). Therefore, the vertical asymptotes are \( \theta = 0 \) and \( \theta = n\pi, n\neq 0 \).

Key Concepts

Understanding Rational Functions

Rational functions play a significant role in calculus and algebra. They are expressions that can be written as the quotient of two polynomials. Typically, a rational function is defined as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial expressions, and \( Q(x) eq 0 \). If the denominator \( Q(x) \) equals zero, then the rational function is undefined at that value of \( x \).
These undefined points are often the locations of vertical asymptotes, if the numerator \( P(x) \) doesn't also become zero at those points. Identifying the vertical asymptotes involves solving \( Q(x) = 0 \) and examining if \( P(x) \) also equals zero at those points.
If the numerator is non-zero at these points, then a vertical asymptote exists. Rational functions like \( g(\theta) = \frac{\tan \theta}{\theta} \) require careful consideration of the zeros in their denominators, as these are key to finding vertical asymptotes.

Exploring Trigonometric Functions

Trigonometric functions, such as \( \tan \theta \), are fundamental in mathematics, often linked to angles and periodic behaviors. The tangent function \( \tan \theta \) is interesting due to its inherent properties: it is undefined for values of \( \theta \) where \( \cos \theta = 0 \), which occur at \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.
This undefined nature translates directly into potential vertical asymptotes when combined with other expressions, like in the case of the function \( g(\theta) = \frac{\tan \theta}{\theta} \). Here, understanding both the periodic nature of \( \tan \theta \) and its undefined points is crucial in pinpointing where vertical asymptotes occur.
The trigonometric functions' graph behaviors, especially the tangent, often result in sharp turns and breaks that denote these asymptotes, distinguishing them from continuous graphs.

Comprehending Undefined Values

In mathematics, a function becomes undefined at points where its expression does not have a valid numerical value. This usually arises when dividing by zero, as is common in rational functions.
For \( g(\theta) = \frac{\tan \theta}{\theta} \), the expression is undefined at \( \theta = 0 \) because \( \theta \) is in the denominator, leading to division by zero. Additionally, \( \tan \theta \) is also undefined at \( \theta = n\pi, neq 0 \), as the periodic zeroes of sine influence this.
Understanding undefined values is essential for determining where vertical asymptotes occur in a function. It involves identifying where each element of the function's expression becomes problematic, especially in terms of division by zero, which is central in causing these vertical asymptotes.