Problem 55 - Calculus

Question

In Problems \(52-57\), give an example of: A function that has a vertical asymptote at \(x=-7 \pi\)

Step-by-Step Solution

Answer

\( f(x) = \frac{1}{x + 7\pi} \) has a vertical asymptote at \( x = -7\pi \).

Step 1: Understand Vertical Asymptote

A vertical asymptote is a line where a function's value approaches infinity as the input approaches a certain value. For a function \( f(x) \), a vertical asymptote at \( x = a \) typically occurs when the function has a factor in the denominator that is zero when \( x = a \) and does not have a corresponding zero in the numerator.

Step 2: Formulate the Denominator

To create a vertical asymptote at \( x = -7\pi \), the denominator of the function must be zero at \( x = -7\pi \). This can be achieved by a factor of \( (x + 7\pi) \) in the denominator. The simplest form for this is \( \frac{1}{x + 7\pi} \).

Step 3: Verify Function Behavior

To confirm the vertical asymptote, consider the function \( f(x) = \frac{1}{x + 7\pi} \). As \( x \to -7\pi^+ \), \( f(x) \to +\infty \), and as \( x \to -7\pi^- \), \( f(x) \to -\infty \), indicating a vertical asymptote at \( x = -7\pi \).

Key Concepts

Asymptote Behavior

Understanding asymptote behavior is crucial when studying functions with vertical asymptotes. A vertical asymptote represents a value of the variable, commonly denoted as \( x = a \), where the function \( f(x) \) becomes unbounded or undefined. As the input value approaches \( x = a \), either from the left or right, the function's value skyrockets towards positive or negative infinity.
This means the function doesn't touch or cross this vertical line, it just gets closer and closer. Asymptotes are essential in describing how a function behaves as it gets near certain critical points, offering insights into the function's overall structure.
For instance, if we consider the function \( f(x) = \frac{1}{x + 7\pi} \), its behavior changes dramatically as \( x \) nears \( -7\pi \). Thus, the vertical asymptote at \( x = -7\pi \) captures this rapid change in behavior.
Remember, recognizing the presence of a vertical asymptote allows us to predict and describe the nature of the function around certain points efficiently.

Function Denominator

The function denominator plays a pivotal role in determining where vertical asymptotes occur. Essentially, vertical asymptotes are formed where the denominator of a function equals zero,
as long as the numerator of the function isn't also zero at that point, which would imply a hole rather than an asymptote.
To craft a function that includes a vertical asymptote at a given value, say \( x = -7\pi \), the denominator needs to zero out at that specific \( x \).
By introducing a factor like \( (x + 7\pi) \) in the denominator, as seen in the function \( f(x) = \frac{1}{x + 7\pi} \), we enforce this condition.
Thus, at \( x = -7\pi \), the denominator is zero, ensuring an asymptotic behavior since \( \frac{1}{0} \) is undefined and causes the function's output to diverge to infinity in either direction.
The denominator helps specify the exact location of vertical asymptotes, rendering it a key factor in mapping them out.

Limit Approach

The concept of a limit approach is foundational for defining and understanding vertical asymptotes in functions. When we mention that a function approaches a vertical asymptote,
we are essentially discussing the behavior of the function as \( x \) approaches a particular value where the limit does not exist in the traditional sense.
Take \( f(x) = \frac{1}{x + 7\pi} \) as an example. As \( x \) approaches \( -7\pi \) from the right (notated as \( x \to -7\pi^+ \)), the function's value heads towards positive infinity.
Conversely, if \( x \) approaches from the left (\( x \to -7\pi^- \)), the function heads towards negative infinity. These directional approaches illustrate how the function diverges swiftly from these values, hence confirming the vertical asymptote.
In essence, evaluating the limits from both sides reveals an infinite discontinuity at the asymptote, solidifying its place within the larger scope of function behavior analysis.
This method of using limits to explore how functions behave near specific points helps students predict and interpret such phenomena accurately.