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Rational functions are mathematical expressions representing the division of two polynomials. The form is usually represented as \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). When working with rational functions, identifying vertical asymptotes and holes are crucial steps in understanding their graphical behavior.

In the case of the exercise \( g(x) = \frac{x+3}{x(x-3)} \), we have a rational function where the numerator \( P(x) = x+3 \) and the denominator \( Q(x) = x(x-3) \). The value of \( x \) that makes the denominator zero is of special interest, as these values are potential vertical asymptotes or holes in the graph. It's important for students to realize that not all values resulting in a zero denominator are asymptotes—some may be holes if the corresponding factor is also present in the numerator and cancels out.

Holes occur in the graph of a rational function when there is a common factor in the numerator and the denominator that can be canceled out. These points are not part of the function's graph because they are undefined, even though they may not be visible when plotting the graph.

To look for holes, as in step 2 of the solution, one should factorize both the numerator and the denominator of the function and check for any common factors. If a common factor exists, it will be canceled out, leaving a 'hole' in the graph at that particular \( x \)-value. Since the function \( g(x) = \frac{x+3}{x(x-3)} \) given in the exercise does not have common factors in the numerator and denominator, no holes are present in its graph. This is an important concept for students to master, as holes represent a subtle nuance of rational function behavior.

Understanding asymptotic behavior is essential when analyzing the graph of rational functions. A vertical asymptote occurs at a value of \( x \) where the function grows without bound as \( x \) approaches the asymptote value from either side. Conversely, a horizontal or oblique asymptote describes the behavior of a graph as \( x \) approaches infinity or negative infinity.

In the context of our exercise, step 1 of the solution explains the process of finding vertical asymptotes by setting the denominator to zero and solving for \( x \). The resulting values, in this case \( x=0 \) and \( x=3 \), are the vertical asymptotes, since the function approaches infinity or negative infinity as it gets closer to these \( x \) values. Asymptotic behavior greatly influences the overall shape of a rational function's graph, and students should be encouraged to analyze this trend to better understand how functions behave near these critical values.