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Graphing functions in calculus involves plotting a curve to visually understand and analyze mathematical expressions. By graphically representing functions, you can intuitively see how the function behaves across different values of x. For the function \[y = \frac{2x^{2}-4.5}{x-1.5}\]you will notice that plugging in the graphing calculator helps to reveal insights that might not be immediately obvious from the equation alone.To effectively graph this function, here are a few steps:

• Select a suitable range that covers the area of interest (in this case, around \(x = 1.5\)).
• Observe the curve of the function as well as any special features such as peaks, troughs, or breaks.
• Ensure that the graphing window is narrow around \(x = 1.5\) for finer details.

By graphing, you can visually confirm calculations and better understand the dynamics of the function. This approach helps in identifying areas like discontinuities or points of interest.

When discussing discontinuity within calculus, we're referring to points where a function is not continuous. In simpler terms, it's where the function has breaks, jumps, or holes. For the function \[y = \frac{2x^{2}-4.5}{x-1.5}\]there is a discontinuity at \(x = 1.5\) because the denominator becomes zero. As division by zero is undefined in mathematics, the function cannot have a value at that point.When graphing, this will typically manifest as a hole in the curve at \(x = 1.5\), indicating the function is not defined there. Understanding discontinuities:

• Create awareness of undefined points which can affect domain.
• Identify these points by analyzing the denominator of rational functions.
• Use graphical analysis to find and verify these points quickly.

Recognizing and understanding the nature of discontinuities is crucial for comprehensive calculus insights.

The behavior of a function as it approaches a specific point provides valuable information about limits. In calculus, you're often interested in what a function's value approaches, even if it's not defined at that point.For our focus here, with\[\lim_{x \rightarrow 1.5} \frac{2x^{2}-4.5}{x-1.5}\]we examine the function's behavior closely as \(x\) gets near \(1.5\) from both sides. Using tools like TRACE or TABLE on graphing devices can help observe how the function values (y-values) behave just before and after \(1.5\).Insights from analyzing behavior:

• Observe whether y-values tend to stabilize around a single point as \(x\) nears \(1.5\).
• Identify the limit by noting the y-value it approaches, indicating a consistent result.
• Recognize that, though the function isn't defined at the precise point, the approaching values provide crucial insight about its trend.

By evaluating how the function behaves near \(x = 1.5\), students can successfully determine the limit, which is a central concept in calculus.