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Rational functions are mathematical expressions that represent the ratio of two polynomials. They take on the form \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are both polynomials. These functions can exhibit interesting behavior as the input value, \( x \), becomes very large or very small, leading to the concept of asymptotes, which are lines that the graph of the function can approach but never actually touch.

In the example provided in the exercise, \( f(x) = \frac{2x}{3x^{2}-5} \) is a rational function because it is composed of a polynomial in the numerator, \( 2x \), and a polynomial in the denominator, \( 3x^{2}-5 \). Understanding the behavior of these functions as \( x \) approaches infinity or negative infinity can provide insight into their long-term behavior and help predict horizontal and vertical asymptotes.

The degree of a polynomial is the highest power of the variable, \( x \), that appears in the polynomial with a non-zero coefficient. When looking at a polynomial, such as \( 4x^3 - 2x^2 + x \), we identify the term with the highest exponent, which in this case is \( 4x^3 \), making the degree 3.

The concept of the degree is crucial in finding horizontal asymptotes of rational functions. If we compare the degrees of the numerator and the denominator of a rational function, we can determine its end behavior and whether or not a horizontal asymptote exists. For instance, if the numerator's degree is less than the denominator's degree, as seen in the exercise with \( f(x) = \frac{2x}{3x^{2}-5} \) where the numerator is of degree 1 and the denominator of degree 2, this implies that the horizontal asymptote is the x-axis, or \( y=0 \).

Asymptotic behavior of functions refers to the behavior of a graph as it approaches a line or a point, either along the x-axis or y-axis, to infinity or negative infinity. Horizontal asymptotes are a type of asymptote that shows us the behavior of a function as \( x \) approaches positive or negative infinity.

When we talk about horizontal asymptotes in the context of rational functions, we are often interested in what happens to the \( y \) value as \( x \) grows without bound. For the function \( f(x) = \frac{2x}{3x^{2}-5} \), the fact that the degree of the denominator is greater than that of the numerator informs us that the \( y \) values will get closer and closer to zero as \( x \) increases or decreases. This is due to the higher power of \( x \) in the denominator causing the values of \( f(x) \) to diminish towards zero, thereby identifying \( y=0 \) as the horizontal asymptote.