Problem 273 - Calculus Volume 1

Question

For the following exercises, find the horizontal and vertical asymptotes. $$ f(x)=\frac{x^{3}}{4-x^{2}} $$

Step-by-Step Solution

Answer

Vertical asymptotes: \(x = 2\) and \(x = -2\); No horizontal asymptote.

Step 1: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, and the numerator is not zero. Set the denominator equal to zero and solve for \(x\). For the function \(f(x) = \frac{x^{3}}{4-x^{2}}\), the denominator is \(4-x^{2}\). Set this equal to zero: \(4-x^{2} = 0\). Solve for \(x\): \(x^{2} = 4\), so \(x = \pm 2\). Thus, the vertical asymptotes are at \(x = 2\) and \(x = -2\).

Step 2: Determine Horizontal Asymptotes

Horizontal asymptotes are found based on the degrees of the polynomial in the numerator and denominator. For \(f(x) = \frac{x^{3}}{4-x^{2}}\), the degree of the numerator is 3 and the degree of the denominator is 2. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a rational function where the denominator equals zero, but the numerator does not become zero at the same point. This creates a line that the graph will never touch, providing symmetry and direction in the graph's shape.
To find vertical asymptotes in the function, set the denominator equal to zero and solve for the variable. For example, in the function \( f(x) = \frac{x^{3}}{4-x^{2}} \), the denominator is \( 4 - x^{2} \).

Setting it to zero gives:

• \( 4 - x^{2} = 0 \)
• \( x^{2} = 4 \)
• \( x = \pm 2 \)

These calculations indicate vertical asymptotes at \( x = 2 \) and \( x = -2 \). At these x-values, the function is undefined because the denominator equals zero. As the graph approaches these lines from either direction, it will climb indefinitely towards positive or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes represent the behavior of a function as the input values grow larger and larger, either positively or negatively. They are determined by comparing the degrees of the polynomials in the numerator and the denominator of a rational function.
Essentially, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis (\( y = 0 \)).
Conversely, if the degrees are equal, the leading coefficients of the numerator and denominator tell us the horizontal asymptote. If the degree of the numerator is greater, there is no horizontal asymptote.

For \( f(x) = \frac{x^{3}}{4-x^{2}} \), the numerator's degree (3) is greater than that of the denominator (2), confirming there is no horizontal asymptote for this function.

Polynomials

Polynomials are mathematical expressions involving variables raised to whole-number exponents and coefficients. They form the foundation for much of algebra and calculus. A polynomial's degree is determined by the highest exponent on its variable, influencing its shape and behavior.
In rational functions, the comparison of polynomial degrees in the numerator and denominator determines characteristics like asymptotes.

Looking at the example \( f(x) = \frac{x^{3}}{4-x^{2}} \), the polynomial numerator is \( x^{3} \), making it a cubic polynomial, while the denominator \( 4 - x^{2} \) is quadratic.
Understanding polynomials' structure aids in identifying asymptotic behavior because it explains important aspects such as end behavior, possible intercepts, and curvatures in the graph.