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

Explain your reasoning. Answer: No, a rational function F(x) = f(x) / g(x) does not always have a vertical asymptote at x = a just because g(a) = 0. The existence of a vertical asymptote at x = a depends on the behavior of both the numerator f(x) and the denominator g(x) as x approaches a. Further analysis of f(x) and g(x) and the limit as x approaches a would be required to definitively determine the existence of a vertical asymptote.

Step-by-step solution

Understand the vertical asymptote condition.

A function has a vertical asymptote at x = a if the limit of the function tends to infinity (or negative infinity) as x approaches a. Informally, this means that the function gets very large (either positively or negatively) as we get closer to x = a. Mathematically, we can write this as: $$ \lim_{x \to a} F(x) = \infty \quad \mathrm{or} \quad \lim_{x \to a} F(x) = -\infty. $$

Examine the behavior of F(x) as x approaches a.

Since F(x) = f(x) / g(x), we can rewrite the limit as: $$ \lim_{x \to a} \frac{f(x)}{g(x)}. $$ We know that g(a) = 0, so as x approaches a, g(x) approaches 0.

Investigate the cases and determine if a vertical asymptote exists.

In order to have a vertical asymptote at x = a, the limit must tend to either infinity or negative infinity as x approaches a. There are three main cases that we can encounter: 1. f(a) ≠ 0 and g(a) = 0: In this case, as x approaches a, the numerator f(x) does not approach 0, while the denominator g(x) approaches 0. This will often result in F(x) tending to infinity or negative infinity, creating a vertical asymptote at x = a. 2. f(a) = 0 and g(a) = 0: In this case, as x approaches a, both the numerator and the denominator approach 0. This is known as an indeterminate form, 0/0, and we cannot conclude if F(x) has a vertical asymptote at x = a or not without further analysis. It would be necessary to explore the behavior of f(x) and g(x) more deeply, possibly using L'Hospital's rule, to determine the existence of a vertical asymptote. 3. f(a) is undefined: This case occurs when a is not in the domain of f(x). In this case, it is not possible to evaluate the limit as x approaches a, and we cannot conclude from this information whether or not F(x) has a vertical asymptote at x = a.

Conclusion.

The function F(x) = f(x) / g(x) does not necessarily have a vertical asymptote at x = a just because g(a) = 0. The existence of a vertical asymptote at x = a depends on the behavior of both the numerator f(x) and the denominator g(x) as x approaches a. Further analysis of f(x) and g(x) and the limit as x approaches a would be required to definitively determine the existence of a vertical asymptote.