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Vertical asymptotes are vertical lines that a graph approaches but never touches or crosses. They occur at the values of \(x\) which make the denominator of a rational function zero, provided these values do not also make the numerator zero. Therefore, finding these values is as simple as solving the equation where the denominator equals zero.

For example, with the function \(f(x) = \frac{x-2}{x^{2}-5x+6}\), the denominator is \(x^2 - 5x + 6\). By setting this equation equal to zero and factoring it as \((x-2)(x-3) = 0\), we find that the roots \(x = 2\) and \(x = 3\) are the solutions. These values of \(x\) lead to division by zero, indicating vertical asymptotes at these points on the graph.

Understanding vertical asymptotes is crucial because they represent the points on a graph where the rational function increases or decreases without bound. Knowing if and where these occur can provide insights into the behavior of the graph around these critical values.

Horizontal asymptotes describe the end behavior of a rational function, particularly as \(x\) approaches positive or negative infinity. They reflect where the graph seems to settle or level off horizontally. A simple way to find them is by comparing the degrees of the numerator and the denominator.

For the function \(f(x) = \frac{x-2}{x^{2}-5x+6}\), the degree of the numerator (which is one) is less than the degree of the denominator (which is two). This difference suggests that as \(x\) approaches both \( \infty \) and \(-\infty\), \(f(x)\) approaches zero. Thus, there is a horizontal asymptote at \(y = 0\).

Horizontal asymptotes provide valuable information on the limiting behavior of the function. They help predict where the function values stabilize for large absolute values of \(x\), guiding us in sketching the function accurately.

Asymptotic behavior captures how a function behaves as it approaches certain lines, called asymptotes. For rational functions like \(f(x) = \frac{x-2}{x^{2}-5x+6}\), identifying this behavior involves both vertical and horizontal asymptotes.

Vertical asymptotes at \(x = 2\) and \(x = 3\) indicate where the function will extend infinitely upwards or downwards as \(x\) gets close to these values, while the horizontal asymptote at \(y = 0\) describes how the function behaves as \(x\) grows very large or very small.

This asymptotic behavior provides a complete picture of the function's trajectory. It helps map out the areas where the graph stretches towards infinity and where it flattens out. Understanding these aspects allows us to sketch a more accurate curve and align it closer to the real function, adhering to asymptotic guidelines.