Problem 20 - Calculus of a Single Variable

Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ g(x)=\frac{x^{3}-8}{x-2} $$

Step-by-Step Solution

Answer

The function \(g(x) = \frac{x^3 - 8}{x - 2}\) does not have a vertical asymptote.

Step 1: Finding Roots of the Denominator

Begin by finding the roots of the denominator \(x - 2 = 0\). Solving this equation for x gives \(x = 2\), which is the root of the denominator.

Step 2: Checking the Numerator

Now, substitute \(x = 2\) into the numerator. If the numerator does not equal zero then we can say that a vertical asymptote exists at this point. Plugging in \(x = 2\) gives \(2^3 - 8 = 0\). The resulting value for the numerator indeed equals zero.

Step 3: Determining the Vertical Asymptote

Since both the numerator and denominator are zero at x=2, we have a hole in the graph rather than a vertical asymptote. As a result, the graph of the function \(g(x)\) does not have a vertical asymptote.

Key Concepts

Limits in Calculus

When exploring the behavior of functions within calculus, the concept of limits is fundamental. A limit can be seen as the value that a function approaches as the input (or x-value) gets closer to a certain point. It represents what the function is 'trying to reach' at this point, even if it doesn't necessarily get there.
In the context of finding vertical asymptotes, limits help us determine what happens as we approach certain critical x-values from both the left and the right side. A vertical asymptote occurs when the value of a function increases without bound (it goes off to infinity) or decreases without bound (it goes down to negative infinity) as the input approaches a particular value.

• To check for vertical asymptotes, we typically look at the points where the denominator of a rational function is zero.
• If the function's limit approaches infinity or negative infinity as x approaches this critical value from either side, then we have a vertical asymptote there.

It's essential not just to find where the denominator is zero but to also explore the behavior of the function around these points using limits to confirm the presence of a vertical asymptote.

Rational Functions

Rational functions are ratios of two polynomials. They are represented as \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials with the condition that \(Q(x)\) should not be zero, as division by zero is undefined.
In the exercise given, \(g(x)=\frac{x^{3}-8}{x-2}\) is a rational function with its numerator \(x^{3}-8\) and denominator \(x-2\). The zeros of the denominator—which can be potential vertical asymptotes—are particularly important in the analysis of these functions.

• If the denominator has a root (zero point) that is not canceled out by a root in the numerator, the function typically has a vertical asymptote there.
• If a root in the denominator is also a root in the numerator (as in this exercise), it may result in a hole rather than a vertical asymptote, depending on the multiplicity of the roots.

The process of identifying these critical points and understanding their implications on the graph of the function is an integral part of working with rational functions.

Asymptote Analysis

Asymptote analysis involves identifying lines that a function's graph approaches but never actually touches. Typically, there are two types of asymptotes: vertical and horizontal. Vertical asymptotes occur at values of x where the function goes to infinity, while horizontal asymptotes occur as x tends to infinity, and the function values approach a constant value.
For the exercise under discussion:

Finding Potential Vertical Asymptotes

We begin by identifying points where the denominator equals zero, as the function cannot exist at these points. These are our potential vertical asymptotes.

Checking Numerator Values

We then examine the numerator at these points. If the numerator is non-zero where the denominator is zero, the function approaches infinity, confirming a vertical asymptote. However, if both the numerator and denominator are zero, the function does not necessarily have an asymptote; instead, it might have a hole, based on whether the zero is a simple root or a higher multiplicity root.
Asymptote analysis requires careful consideration of both the numerator and denominator of rational functions. Understanding this will help students avoid misconceptions and correctly determine the behavior of functions.