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A vertical asymptote occurs in a rational function when the value of the denominator is zero, specifically where the fraction itself becomes undefined. To identify these points, we set the denominator of the rational function equal to zero and solve for the variable.

In the function \( f(x)=\frac{1}{(x-2)^{3}} \), we are dealing with a rational function where the numerator is a constant (1) and the denominator is \((x-2)^3\). Clearly, the expression \((x-2)^3=0\) when \(x=2\).

Consequently:

• There is a vertical asymptote at \(x=2\).
• The graph of the rational function will approach, but never touch or cross, this vertical line.

This behavior signals that as \( x \) gets extremely close to 2, the function \( f(x) \) will tend towards positive or negative infinity, creating a sharp climb or drop on the graph.

Horizontal asymptotes in rational functions indicate the value that a function approaches as the input (\(x\)) becomes very large or very small, essentially describing the end behavior of the function.

For \( f(x)=\frac{1}{(x-2)^3} \), the degrees of the numerator and denominator determine the horizontal asymptote:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote will be \(y=0\).
• If both degrees are equal, the asymptote would be the ratio of the leading coefficients.
• If the numerator's degree is greater, there is no horizontal asymptote - but instead, an oblique one might exist.

Since the numerator's degree is 0 (constant) and the denominator's degree is 3, the horizontal asymptote for \(f(x)\) lies along \(y=0\). As \(x\) approaches both infinity and minus infinity, \(f(x)\) will settle around 0, but never reaching it exactly, creating a flat line in the horizontal direction on the graph.

Rational functions are fractions where both the numerator and the denominator are polynomials. These types of functions are prominent in calculus and algebra as they model relationships where quantities are inversely related.

They can create unique graph behavior including vertical and horizontal asymptotes, holes, and sometimes oblique asymptotes. Key characteristics of rational functions include:

• Vertical asymptotes occurring where the denominator equals zero, leading to undefined values in the function.
• Horizontal or oblique asymptotes that describe the end behavior as \(x\) grows larger in both positive or negative directions.

For instance, in the function \( f(x)=\frac{1}{(x-2)^3} \), the constant numerator and the cubic denominator together determine the nature and position of the asymptotes. This setup exemplifies a typical rational function where the balance of the polynomial degrees dictates the asymptotic behavior.