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Vertical asymptotes are key features of the graph of a function. They occur at values of \( x \) that make the denominator of a rational function zero, as the function's value approaches infinity in magnitude. For the function \( f(x) = \frac{8}{x^2 - 9} \), we find vertical asymptotes by setting the denominator equal to zero: \( x^2 - 9 = 0 \). Solve this to get \( x = \pm 3 \). These asymptotes are important to include in a graphing window as they mark where the function will appear to "shoot off" towards infinity. Understanding where vertical asymptotes are helps in correctly identifying the behavior of the graph in sections close to these critical points.

Horizontal asymptotes represent the behavior of a function as \( x \) approaches infinity or negative infinity. For \( f(x) = \frac{8}{x^2-9} \), as \( x \to \pm \infty \), the term \( x^2 \) becomes significantly larger than 9. This makes the function approach zero, creating a horizontal asymptote at \( y = 0 \). This indicates that the graph will get closer and closer to the x-axis, but never actually touches it, as \( x \to \pm \infty \). Recognizing horizontal asymptotes helps in understanding the "flattening" behavior of the graph as it stretches out wide, but ultimately tending towards a steady, limiting value.

Examining end behavior provides insight into what happens to the values of \( f(x) \) as \( x \) goes toward positive or negative infinity. Knowing that the function \( f(x) = \frac{8}{x^2 - 9} \) has its horizontal asymptote at \( y = 0 \), we can say that as \( x \to \infty \), \( f(x) \to 0 \), and similarly as \( x \to -\infty \), \( f(x) \to 0 \). This indicates that regardless of how large \( |x| \) becomes, the function value will get closer and closer to zero. This end behavior is crucial for the accurate portrayal of the overall function's graph, showing off the diminishing returns of \( f(x) \) over vast \( x \) values.

Selecting an appropriate graphing window is pivotal in showcasing the primary characteristics of a function. For \( f(x) = \frac{8}{x^2 - 9} \), we need to ensure our window captures all important behaviors including the vertical and horizontal asymptotes. In this case, choosing an \( x \) range from \(-10\) to \(10\) ensures we cover the vertical asymptotes at \( x = -3 \) and \( x = 3 \). The \( y \) range of \(-10\) to \(10\) accommodates the fluctuations around the horizontal asymptote at \( y = 0 \). This window effectively displays the overall graphical picture, capturing the sharp rises and declines near vertical asymptotes and the leveling off toward the horizontal asymptote.

Understanding limits at infinity involves exploring the value to which \( f(x) \) approaches as \( x \) becomes infinitely large or small. For the function \( f(x) = \frac{8}{x^2 - 9} \), as \( x \to \infty \) or \( x \to -\infty \), the term \( \frac{8}{x^2} \) dominates the behavior, trending the function towards zero. This calculation confirms the presence of a horizontal asymptote at \( y = 0 \). Knowing these limits helps students anticipate how the function behaves in its extended domains and visualize how it stretches out nearly parallel to the x-axis without crossing it, which is vital for an accurate graph representation.