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Rational functions are expressions found in the form of a fraction, where both the numerator and the denominator are polynomials. They can be written as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero. This structure allows rational functions to illustrate many different behaviors, including asymptotic behavior.
Rational functions can have diverse characteristics depending on the degree of the polynomials involved. For instance, if both the numerator and denominator polynomials are of equal degree, it plays an important role in determining the shape of the function on a graph, particularly when identifying horizontal asymptotes.
Such functions are widely used in mathematical modeling since they can represent situations where one quantity varies with respect to another, such as rates of growth or decline, and have applications across a range of fields from physics to economics.

Vertical asymptotes are lines that a graph approaches but never actually touches or crosses as it moves along the x-axis. They occur where the rational function's denominator equals zero, causing the function to be undefined.
To find vertical asymptotes, set the denominator of the rational function to zero and solve for \(x\). For example, with a function \( t(x)=\frac{x^{2}}{x^{2}-x-6} \), we derive the conditions for vertical asymptotes by solving the equation \( x^2 - x - 6 = 0 \). Factoring gives us \((x - 3)(x + 2) = 0\), leading to \( x = 3 \) and \( x = -2 \) as solutions.
These values are where vertical asymptotes occur, provided they do not cancel out with any factors in the numerator. Vertical asymptotes signify points of abrupt change or infinite limits, greatly shaping the overall behavior of the graph.

Horizontal asymptotes are lines that a graph of a function approaches as \(x\) goes to positive or negative infinity. For rational functions, they can be determined by the degrees of the numerator and denominator.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). If the degrees are the same, as in \(t(x)=\frac{x^{2}}{x^{2}-x-6}\), we divide the leading coefficients of the numerator and denominator, resulting in a horizontal asymptote \(y = 1\) because the degrees are both 2, and the leading coefficients are both 1.
This tells us about the end behavior of the function, indicating that as the inputs become very large or very small, the outputs will level off toward the asymptote's value, guiding the shape of the graph as it stretches out infinitely.