Problem 32 - Precalculus Mathematics for Calculus

Question

Find all horizontal and vertical asymptotes (if any). $$r(x)=\frac{x^{3}+3 x^{2}}{x^{2}-4}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = 2\) and \(x = -2\); no horizontal asymptotes.

Step 1: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function is equal to zero, as long as the numerator does not also equal zero there. To find the points where vertical asymptotes occur, solve the equation \(x^2 - 4 = 0\):\[x^2 - 4 = 0\]\[x^2 = 4\] \[x = 2\] or \[x = -2\]. So, the vertical asymptotes are at \(x = 2\) and \(x = -2\).

Step 2: Determine Horizontal Asymptotes

Horizontal asymptotes are found by comparing the degrees of the polynomial in the numerator and denominator. Here, the polynomial degree of the numerator \(x^3 + 3x^2\) is 3, while the degree of the denominator \(x^2 - 4\) is 2. Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote. Instead, the graph potentially has an oblique asymptote, but no horizontal asymptote.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are a key feature of rational functions, which are functions expressed as the ratio of two polynomials. A vertical asymptote occurs where the function approaches infinity, typically where the denominator is zero and the numerator is non-zero. This is because division by zero is undefined, causing the function's value to spike towards ±∞.

In the function given, \( r(x)=\frac{x^{3}+3x^{2}}{x^{2}-4} \), to find the vertical asymptotes, we need to set the denominator equal to zero and solve for \( x \).

\[ x^2 - 4 = 0\]
Solving this equation, we get:
\[ x^2 = 4 \]
\[ x = \pm 2\]

This process identifies two vertical asymptotes at \( x = 2 \) and \( x = -2 \). Note, it is crucial that the numerator does not also equal zero at these points, as this would instead result in a hole in the graph rather than a vertical asymptote.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) approaches either positive or negative infinity. They indicate the value that the function is tending towards but never actually reaches.

For the original function, \( r(x)=\frac{x^{3}+3x^{2}}{x^{2}-4} \), we analyze the degrees of the polynomials in both the numerator and the denominator to determine horizontal asymptotes:

• Numerator degree: 3 (due to \( x^3 \))
• Denominator degree: 2 (due to \( x^2 \))

When the degree of the numerator is greater than the degree of the denominator, as it is here, there is no horizontal asymptote. Instead, the graph may contain an oblique asymptote, which is a diagonal line that the graph approaches but does not intersect. Horizontal asymptotes are strictly horizontal lines approaching finite values, and with these degree relationships, it's not applicable.

Rational Functions

Rational functions are expressed as the ratio of two polynomial functions. Their defining feature is the polynomial in the numerator and the denominator. These functions can display unique asymptotic behavior, such as vertical and horizontal asymptotes.

Studying rational functions involves looking at the degrees of these polynomials. The degree is the highest power of \( x \) in the polynomial expression. The comparison of these degrees helps in determining the asymptotic properties of the function:

• If the degree of the numerator is less than that of the denominator, there is a horizontal asymptote at \( y = 0 \).
• If the degrees are equal, the horizontal asymptote is at \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator, respectively.
• If the numerator's degree is greater, like in our given function, no horizontal asymptote exists, suggesting possible oblique asymptotes.

Understanding rational functions and their asymptotes aids in graphing these functions and predicting their behavior across their domains.