Problem 38 - Calculus

Question

In Problems \(36-38,\) find all horizontal and vertical asymptotes for each rational function. $$f(x)=\frac{5 x^{3}+7 x-1}{x^{3}-27}$$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = 3\); horizontal asymptote at \(y = 5\).

Step 1: Understanding Asymptotes

In this exercise, we need to find the horizontal and vertical asymptotes of the rational function \(f(x) = \frac{5x^3 + 7x - 1}{x^3 - 27}\). Asymptotes are lines that the graph of the function approaches but never touches as \(x\) goes to infinity or some finite value.

Step 2: Finding Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. We set the denominator equal to zero: \(x^3 - 27 = 0\). Solving this, we find \(x^3 = 27\) or \(x = 3\). Since the numerator is not zero when \(x = 3\), there is a vertical asymptote at \(x = 3\).

Step 3: Determining Horizontal Asymptotes

Horizontal asymptotes depend on the degrees of the numerator and denominator. Here, both are degree 3. When degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator is 5 and the leading coefficient of the denominator is 1. Thus, the horizontal asymptote is \(y = \frac{5}{1} = 5\).

Step 4: Conclusion on Asymptotes

The function \(f(x) = \frac{5x^3 + 7x - 1}{x^3 - 27}\) has a vertical asymptote at \(x = 3\) and a horizontal asymptote at \(y = 5\).

Key Concepts

Rational Functions

Rational functions are a type of function defined as the ratio of two polynomials. In a rational function, the numerator and the denominator are both polynomial expressions. A simple example is given by

• Numerator: A polynomial like \( P(x) = 5x^3 + 7x - 1 \)
• Denominator: Another polynomial such as \( Q(x) = x^3 - 27 \)

Thus, our rational function becomes \( f(x) = \frac{P(x)}{Q(x)} \).
It's essential to understand that the domain of a rational function includes all real numbers except where the denominator is zero. Such values cause the function to be undefined, leading to vertical asymptotes.
Rational functions are unique because their graphs can exhibit interesting behaviors like having asymptotes, which act as boundaries that the function cannot cross.

Vertical Asymptotes

Vertical asymptotes arise in rational functions at points where the denominator equals zero and the numerator does not equal zero simultaneously. This concept is essential because it tells us where a function is undefined and how it behaves near those points.
For the given function \( f(x) = \frac{5x^3 + 7x - 1}{x^3 - 27} \), the vertical asymptote is determined as follows:

• Identify where the denominator \( Q(x) = x^3 - 27 \) equals zero.
• Solve \( x^3 - 27 = 0 \), which gives \( x = 3 \) as the value.

As the numerator \( 5x^3 + 7x - 1 \) is not zero at \( x = 3 \), a vertical asymptote exists at this point. The function's graph will approach but never actually touch or cross this vertical line. It essentially divides the graph, creating distinct behaviors on either side of the line.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as the variable \( x \) approaches either positive or negative infinity. They provide a line that the graph will tend to approach from a distance.

• In rational functions, these depend on the degrees of the polynomials in the numerator and the denominator.

Considering the function \( f(x) = \frac{5x^3 + 7x - 1}{x^3 - 27} \), notice both the numerator and the denominator have a degree of 3.
In such cases, we find the horizontal asymptote by comparing the leading coefficients of these polynomials:

• The leading coefficient of the numerator = 5
• The leading coefficient of the denominator = 1

The horizontal asymptote is thus \( y = \frac{5}{1} = 5 \). As \( x \) grows large in positive or negative directions, the graph of the function will tend to level off at \( y = 5 \). This gives us a horizontal line that the function can't exceed at extreme values of \( x \).