Final Answer:
To use a table to examine the asymptotic behavior of a rational function, you can create a table of values for x and then calculate the corresponding y-values using the rational function. By observing the behavior of the y-values as x approaches positive or negative infinity, you can determine the horizontal and vertical asymptotes of the function.
Step-by-step explanation:
When examining the asymptotic behavior of a rational function, it’s essential to create a table of values for x and then calculate the corresponding y-values using the given rational function. For example, consider the rational function f(x) = (2x² + 3x - 5) / (x - 1). To examine its asymptotic behavior, we can create a table with different x-values such as -10, -5, -1.5, -1.1, -1.01, 0, 1.01, 1.1, 1.5, 5, and 10. Then we calculate the corresponding y-values by substituting these x-values into the function.
By observing the behavior of the calculated y-values as x approaches positive or negative infinity, we can determine the horizontal and vertical asymptotes of the function. In this example, as x approaches positive or negative infinity, we observe that the y-values approach certain constant values or become unbounded. These observed limits help us identify the horizontal and vertical asymptotes of the rational function.
In conclusion, using a table to examine the asymptotic behavior of a rational function involves creating a table of x-values and calculating their corresponding y-values to observe their behavior as x approaches positive or negative infinity. This process aids in identifying the horizontal and vertical asymptotes of the rational function.