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Limits are essential in calculus, helping us understand the behavior of functions as they approach specific values. In our exercise, several limits have been used to describe how the function behaves near certain points or as it moves towards infinity. For instance, when \( \lim_{x \to 2} f(x) = -\infty \), it tells us that as \( x \) gets very close to 2, but not equal to 2, the function value \( f(x) \) decreases without bound.
Similarly, \( \lim_{x \to 0^+} f(x) = \infty \) and \( \lim_{x \to 0^-} f(x) = -\infty \) describe the behavior of \( f(x) \) as it approaches 0 from the positive and negative sides, respectively. These limits inform us that \( f(x) \) rises towards infinity and falls towards negative infinity in these scenarios.
Also, \( \lim_{x \to \infty} f(x) = \infty \) means that as \( x \) becomes very large, \( f(x) \) also becomes very large. Knowing these behaviors helps us understand and predict the function's pattern.

Asymptotes are lines that the graph of a function approaches but never touches. Understanding them is critical when analyzing the graph of a function. In this exercise, we have both vertical and horizontal asymptotes.
A vertical asymptote occurs when a function grows very large or decreases very large near certain points. Here, we have vertical asymptotes at \( x = 0 \) and \( x = 2 \). This means the graph of the function will shoot up to infinity or plunge to negative infinity as \( x \) approaches 0 and 2 respectively.

• Vertical asymptote at \( x = 0 \): As \( x \to 0^+ \), \( f(x) \to \infty \) and as \( x \to 0^- \), \( f(x) \to -\infty \).
• Vertical asymptote at \( x = 2 \): As \( x \to 2 \), \( f(x) \to -\infty \).

There is also a horizontal asymptote. As \( x \to -\infty \), \( f(x) \to 0 \), indicating a horizontal asymptote at \( y = 0 \). This horizontal line represents the function approaching a constant value as \( x \) becomes very negative.
These asymptotes provide a framework for sketching the function's behavior.

Rational functions are fractions where both the numerator and the denominator are polynomials. They are particularly interesting because their division leads to some unique behaviors, such as the presence of asymptotes. The given solution uses the rational function \( f(x) = \frac{x}{(x-2)x^2} \).
This function embodies various characteristics dictated by its form. The denominator includes \( (x-2)x^2 \), which informs us about the function's vertical asymptotes and limits. For instance, as \( x \to 2 \) or \( x \to 0 \), the denominator becomes zero, leading to vertical asymptotes at these values.
The degree of the polynomial in the denominator exceeds that in the numerator when \( x \to \infty \), resulting in behaviors matching those described by the limits specified in the exercise.

Graph sketching is the art of representing a function's behavior visually, incorporating key information about its limits and asymptotes. Through sketching, we support mathematical analysis with a visual framework.

• Start by marking the vertical asymptotes on the graph at \( x = 0 \) and \( x = 2 \). Draw dashed lines to indicate these barriers that the function approaches but never crosses.
• Next, illustrate the horizontal asymptote at \( y = 0 \) as x moves toward \(-\infty\).
• Finally, plot the behavior of the function in these sections, ensuring it spikes towards infinity near \( x = 0^+ \), descends towards \(-\infty\) at \( x = 0^- \), and \( x = 2 \). As \( x \to \infty \), show the graph moving upwards.

By considering both vertical and horizontal asymptotes, limits, and the rational function's form, we accurately sketch the function’s graph, offering a comprehensive, visual view of its mathematical form.