Question

Consider the function \(F(x)=f(x) / g(x)\) with \(g(a)=0 .\) Does \(F\) necessarily have a vertical asymptote at \(x=a ?\) Explain your reasoning.

Short answer

Answer: No, the function \(F(x)\) does not necessarily have a vertical asymptote at \(x=a\) if \(g(a)=0\). It is essential to confirm that the limits as \(x\) approaches \(a\) are infinite. If the limits are found to be infinite, then a vertical asymptote exists at \(x=a\).

Step-by-step solution

Definition of a Vertical Asymptote

In order to determine if function \(F(x)\) has a vertical asymptote at \(x=a\), we need to understand the definition of a vertical asymptote. A vertical asymptote is a vertical line \(x=a\), such that as \(x\) approaches \(a\), the function goes to either positive or negative infinity. Mathematically, a function has a vertical asymptote at \(x=a\) if one of the following is true: 1. \(\lim_{x \to a^-} F(x) = \pm \infty\) 2. \(\lim_{x \to a^+} F(x) = \pm \infty\) Now let's analyze the given function \(F(x)\) to see if it meets these conditions.

Analyze the Behavior of \(F(x)\) as \(x\) Approaches \(a\)

If \(g(a) = 0\) and \(f(a) \neq 0\), then one or both of the limits mentioned above could be infinite. We need to examine the behavior of the function as \(x\) approaches \(a\). 1. If the limits exist and are finite, the function does not have a vertical asymptote at \(x=a\). 2. If the limits are infinite, the function has a vertical asymptote at \(x=a\).

Provide an Example

Let's consider an example to illustrate this analysis. Let \(f(x) = x\) and \(g(x) = (x-1)(x+1)\). The function \(F(x) = \frac{f(x)}{g(x)}\) has \(g(1) = 0\) and \(g(-1)= 0\). Therefore, let's analyze the limits as \(x\) approaches 1 and -1: 1. \(\lim_{x \to 1^-} F(x) = \lim_{x \to 1^-} \frac{x}{(x-1)(x+1)} = - \infty\) 2. \(\lim_{x \to 1^+} F(x) = \lim_{x \to 1^+} \frac{x}{(x-1)(x+1)} = \infty\) 3. \(\lim_{x \to -1^-} F(x) = \lim_{x \to -1^-} \frac{x}{(x-1)(x+1)} = \infty\) 4. \(\lim_{x \to -1^+} F(x) = \lim_{x \to -1^+} \frac{x}{(x-1)(x+1)} = -\infty\) In this example, all four limits are infinite, so the function \(F(x)\) has vertical asymptotes at \(x=1\) and \(x=-1\). Therefore, it is not necessarily true that \(F(x)\) will have a vertical asymptote at \(x=a\), because the limits must be confirmed to be infinite. However, if the limits are found to be infinite, then a vertical asymptote exists at \(x=a\).

Key concepts

Understanding Limits in Functions

When dealing with mathematical functions, understanding limits is crucial. Limits describe how a function behaves as it approaches a certain point. This concept is fundamental in calculus.
When we talk about limits approaching infinity, we are looking at what happens when the input value of a function gets very close to some value, say \(x = a\), but not equal to it. The result can tell us if the function is growing very large positively or negatively.
The notation \(\lim_{{x \to a^+}} F(x) = \pm \infty\) implies that as \(x\) gets closer to \(a\) from the right, the function \(F(x)\) becomes infinitely large or small. Similarly, \(\lim_{{x \to a^-}} F(x) = \pm \infty\) indicates the same behavior from the left side.
Calculating these limits helps us identify peculiar behaviors like vertical asymptotes, which are lines where the function doesn't exist but gets infinitely close to. This analysis is conducted by examining how the numerator and denominator behave as \(x\) approaches \(a\).

Exploring Rational Functions

Rational functions are functions represented by the ratio of two polynomials. They take the form \(\frac{f(x)}{g(x)}\) where \(f(x)\) and \(g(x)\) are polynomial functions. These functions can exhibit interesting behaviors, particularly at points where the denominator is zero, a condition known as a singularity.
Unlike polynomials, rational functions can have asymptotes, which are lines that the graph approaches but never touches. A vertical asymptote occurs at \(x = a\) if the denominator equals zero and cannot be canceled by a common factor in the numerator.
Analyzing the numerator and denominator individually reveals how the function behaves around these singularities. If \(g(a) = 0\) and \(f(a) eq 0\), the function has potential vertical asymptotes since the denominator enforces division by a number approaching zero.
However, it's important to know that a zero denominator doesn't always mean a vertical asymptote. If both the numerator and denominator share the factor making \(g(a)\) zero, the point might instead be a hole or removable discontinuity in the graph of the function.

Behavior Analysis Near Vertical Asymptotes

Behavior analysis around vertical asymptotes offers insights into how functions behave near singular points. To determine if a rational function like \(F(x) = \frac{f(x)}{g(x)}\) has a vertical asymptote at \(x = a\), careful limit calculation is required.
Let's consider the example function \(F(x) = \frac{x}{(x-1)(x+1)}\) to analyze its behavior as \(x\) approaches 1 and -1, where \(g(x)\) becomes zero. Here, boundary approach from both directions should be evaluated:

• \(\lim_{{x \to 1^-}} F(x) \) and \(\lim_{{x \to 1^+}} F(x) \)
• \(\lim_{{x \to -1^-}} F(x) \) and \(\lim_{{x \to -1^+}} F(x) \)

In these calculations, if the limits result in infinity, then \(x = 1\) and \(x = -1\) are points where the function approaches an asymptote. Conversely, finite limits indicate no vertical asymptote.
This behavioral analysis not only indicates the presence of asymptotes but also describes the directional tendencies (towards infinity or negative infinity) of the function values as they approach undefined regions.