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 \(F(x) = \frac{f(x)}{g(x)}\) does not necessarily have a vertical asymptote at \(x = a\) just because \(g(a)=0\). The presence of a vertical asymptote depends on the behavior of both \(f(x)\) and \(g'(x)\) as \(x\) approaches \(a\). If the necessary conditions for a vertical asymptote are satisfied, then \(F(x)\) would have a vertical asymptote at \(x=a\). Otherwise, it does not.

Step-by-step solution

Definition Of A Vertical Asymptote

A vertical asymptote occurs at a point \(x=a\) if the function \(F(x)\) approaches infinity or minus infinity as \(x\) approaches \(a\). In other words, if either \(\lim_{x \to a^+} F(x) = \pm \infty\) or \(\lim_{x \to a^-} F(x) = \pm \infty\), then the function has a vertical asymptote at \(x = a\).

Analyzing The Limit Of The Function Approaching \(x=a\)

For the function \(F(x) = \frac{f(x)}{g(x)}\), we need to analyze the behavior of the limit as \(x\) approaches \(a\). We know that \(g(a)=0\), but in order to determine if \(F(x)\) approaches infinity or minus infinity near \(x=a\), we need to analyze \(\lim_{x \to a} f(x)\) and \(\lim_{x \to a} g'(x)\). This is because, if \(\lim_{x \to a} f(x) \neq 0\) and \(\lim_{x \to a} g'(x) \neq 0\), then the function \(F(x)\) might have a vertical asymptote at \(x=a\). On the other hand, if \(\lim_{x \to a} f(x) = 0\) or \(\lim_{x \to a} g'(x)\) does not exist or equals 0, then the function \(F(x)\) does not have a vertical asymptote at \(x=a\).

Conclusion

The function \(F(x) = \frac{f(x)}{g(x)}\) does not necessarily have a vertical asymptote at \(x=a\) just because \(g(a)=0\). The presence of a vertical asymptote depends on the behavior of both \(f(x)\) and \(g'(x)\) as \(x\) approaches \(a\). If the necessary conditions for a vertical asymptote are satisfied (as mentioned in Step 2), then \(F(x)\) would have a vertical asymptote at \(x=a\). Otherwise, it does not.

Key concepts

Understanding Limits

In calculus, understanding limits is important, especially when exploring issues like vertical asymptotes. A limit examines what happens to a function as the input approaches a certain value. To check if a vertical asymptote exists in a function like \( F(x) = \frac{f(x)}{g(x)} \), we need to consider the limits of both the numerator and the denominator as \( x \) approaches a point \( a \).

Here are some key points about limits in this context:

• If \( \lim_{x \to a} f(x) eq 0 \) and \( \lim_{x \to a} g(x) = 0 \), a vertical asymptote might appear because the denominator approaches zero, leading to a possible infinite value.
• If \( g(x) \) approaches zero while \( f(x) \) does not, the function's value may grow very large positively or negatively.

When examining limits for rational functions, always consider how both parts of the function behave as \( x \) nears \( a \). This determines if a vertical asymptote truly exists.

Exploring Rational Functions

Rational functions are expressions of the form \( \frac{f(x)}{g(x)} \), where \( f(x) \) and \( g(x) \) are polynomials. These functions often present interesting behaviors, especially when it comes to finding asymptotes.

Here's what you should know about rational functions:

• A vertical asymptote in a rational function occurs primarily where the denominator is zero, provided that the numerator isn’t zero at the same point.
• Checking endpoints where \( g(a) = 0 \) is crucial. If \( f(x) \) is not zero as \( x \) approaches \( a \), \( F(x) \) may display a vertical asymptote.
• Closely examine how \( g(x) \) behaves as it gets close to these critical points in order to understand the entire function's behavior.

Rational functions highlight how small changes in \( g(x) \) can lead to significant changes in the overall function, emphasizing the importance of checking the conditions for asymptotes carefully.

Identifying Discontinuous Functions

Discontinuity is a big part of understanding when vertical asymptotes might occur in functions. Discontinuous functions do not have a continuous graph, meaning there are breaks or jumps.

Some points to consider about discontinuity:

• If \( F(x) = \frac{f(x)}{g(x)} \) has a point \( x=a \) where \( g(a) = 0 \) but \( f(a) eq 0 \), expect a discontinuity at that point as the function might approach infinity.
• Discontinuities can occur at any point where the limit of \( F(x) \) doesn’t exist, such as points where function values become unbounded.
• Understanding where a function is discontinuous helps in identifying potential asymptotes by analyzing the limits around those critical points.

Recognizing these gaps or jumps will assist in knowing whether a function will have a vertical asymptote, contributing to a comprehensive understanding of the function's behavior.