91Ó°ÊÓ

A vertical asymptote of a function is a vertical line that a graph approaches but never touches or crosses. This typically occurs when the value of the function becomes unbounded (tending towards infinity) as it nears the line. Vertical asymptotes are most commonly found in rational functions, which are ratios of two polynomials. Here's how you find them:

• Identify the denominator of the function. You set it equal to zero because division by zero is undefined.
• Solve for the variable to find the values that make the denominator zero.
• For the function you have, the denominator is \(3x - 1\). Setting it to zero: \(3x - 1 = 0\) leads to \(x = \frac{1}{3}\).

This tells us that a vertical asymptote exists at \(x = \frac{1}{3}\). Remember, this asymptotic behavior implies that as \(x\) approaches \(\frac{1}{3}\), \(f(x)\) will soar to positive or negative infinity, making this line crucial for understanding the function's long-term behavior near this critical point.

Horizontal asymptotes are horizontal lines that the graph of a function will approach as \(x\) becomes very large (positively or negatively). These lines do not indicate that the function will become infinite, but rather, they show the ceiling or floor that the function's values will never exceed in its tail behavior. Here's how you can determine a horizontal asymptote:

• Examine the degrees of the numerator and the denominator in the rational function.
• If the degree of the numerator is less than the denominator, the horizontal asymptote is \(y = 0\).
• If they are equal, the asymptote is the ratio of the leading coefficients of these terms.
• If the numerator's degree is greater, there is no horizontal asymptote.

For \(f(x)=\frac{x}{3x-1}\), both numerator and denominator have degree 1. The leading coefficient of the numerator (\(x\)) is 1, and for the denominator (\(3x\)), it is 3. Thus, the horizontal asymptote is \(y = \frac{1}{3}\). The function will level out and get closer to this value as \(x\) moves towards positive or negative infinity.

The degree of a polynomial plays a fundamental role in understanding the characteristics of the polynomial functions, particularly in finding asymptotes of rational functions. Here’s how to interpret it:

• The degree of a polynomial in a term is the highest exponent of the variable in the term.
• For example, in \(x^2 + 3x + 7\), the degree is 2, because the highest power of \(x\) is \(x^2\).
• The degree of a polynomial helps in determining the behavior of polynomial functions as \(x\) approaches infinity.

In rational functions such as \(\frac{x}{3x-1}\), understanding the degrees of the numerator and the denominator is key to finding asymptotic behaviors. Here, since both have a degree of 1, the comparison of these two degrees helps you find the horizontal asymptote, as well as indicates where the vertical asymptote might occur. Knowing the degree provides a quick insight into the function's end behavior and aids in sketching its graph.