Question

True or False? In Exercises \(65-68,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is undefined at \(x=0 .\)

Short answer

True. If a function \(f\) has a vertical asymptote at \(x=0,\) then generally \(f\) is undefined at \(x=0.http://x=0.\) Although there might be exceptions, this is typically true for most standard functions.

Step-by-step solution

Understanding Vertical Asymptotes

An asymptote is a line that a curve approaches but never touches. A vertical asymptote represents a value that the function will never reach but will get increasingly close to as the independent variable approaches a specific value. In this case, the function \(f\) will approach but never meet the line \(x=0.\)

Evaluating the Statement

The existence of a vertical asymptote at \(x=0\) is generally indicative of the function \(f\) being undefined at \(x=0\). This is because the function tends to infinity or negative infinity as it gets closer to the asymptote. Therefore, it would be accurate to say that \(f\) is undefined at \(x=0\).

Concluding the Evaluation

Despite a few exceptions, where even though there's an asymptote, the function could still be possibly defined at that point (e.g., removable discontinuities), for most practical and standard functions, if \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is indeed undefined at \(x=0.\)

Key concepts

Undefined Function at Vertical Asymptote

When we encounter a vertical asymptote in a graph of a function, it signifies a sort of 'boundary' that the function will never actually cross. This boundary is represented by a specific x-value, and at this point, the function's behavior dramatically changes - it becomes infinitely large in magnitude, either positively or negatively. This is why we say a function is 'undefined' at a vertical asymptote. However, it's crucial to understand that 'undefined' in this context doesn't mean there's no value whatsoever; rather, it refers to the idea that the function doesn't settle at a finite number.
To further illustrate, consider the function \( f(x) = \frac{1}{x} \). As we approach \( x = 0 \), the values of \( f(x) \) get larger and larger in magnitude, indicating our vertical asymptote. At the point \( x = 0 \), the function does not have a defined numerical value since division by zero is not allowed in mathematics, leading to a conclusion that \( f(x) \) is undefined at this point. This is an ideal example of an undefined function at a vertical asymptote.
Even though undefined, the concept of limits allows us to understand the behavior of \( f(x) \) as it 'approaches' zero, providing insight into the function's growth without requiring an exact value at the asymptote.

Evaluating Vertical Asymptotes

Evaluating vertical asymptotes is a critical skill in understanding the structure of a function. To do this, we look for points in the domain of the function where it becomes undefined, particularly due to division by zero or the logarithm of zero. While graphing, vertical asymptotes appear as lines that the function gets infinitesimally close to but never intersects or touches.
To find these elusive lines, one commonly used technique is to look for values that make the denominator of a rational function equal to zero (since division by zero is undefined) or the argument within a logarithmic function equal to zero. For instance, in a function like \( g(x) = \frac{2}{x-3} \), the vertical asymptote is found at \( x = 3 \) because plugging in 3 for \( x \) gives us a zero in the denominator, making \( g(x) \) undefined. However, it's crucial to check for potential cancellations in the numerator and denominator, which could lead to removable discontinuities rather than asymptotes.
Advanced calculus approaches, such as limits, can help us evaluate what value the function approaches as \( x \) gets near the vertical asymptote, which provides a deeper understanding concerning the behavior of the function around these critical values.

Asymptote Behavior in Functions

The behavior of functions around asymptotes can be intriguing and sometimes complex. As a function approaches a vertical asymptote, the function's values either increase or decrease without bound. This means the outputs of the function become larger and larger (positively or negatively) and do not settle at a finite value. This characteristic, while visually apparent on a graph, also has implications in calculus when discussing limits.
For example, with the function \( h(x) = \frac{5}{(x-1)^2} \), as \( x \) approaches 1, \( h(x) \) grows without bound, indicating a vertical asymptote at \( x=1 \). The function will not just steadily increase or decrease; it will do so at an accelerating rate due to the squared term in the denominator. Interestingly, not all functions exhibit only this classic 'unbounded' behavior. Some functions, like those with a sinusoidal component, might oscillate wildly as they approach the asymptote, adding an extra layer of complexity to their asymptotic behavior.
It's important for students to realize that while asymptotes indicate where a function does not have a finite value, they also guide us in understanding the trend and direction of a function as it moves towards these limits. By studying the function near its vertical asymptotes, we can gain insights into the function's underlying nature and its potential applications.