Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = x³ / (x³ − 8)

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Step 1: Identify the domain of the function y = x³ / (x³ − 8). The domain consists of all real numbers except where the denominator is zero. Set the denominator equal to zero and solve for x: x³ - 8 = 0.

Step 2: Solve the equation x³ - 8 = 0 to find the values of x that are not in the domain. This can be done by factoring or using the cube root: x³ = 8, so x = 2.

Step 3: The domain of the function is all real numbers except x = 2. In interval notation, this is expressed as (-∞, 2) ∪ (2, ∞).

Step 4: To find vertical asymptotes, examine the behavior of the function as x approaches the value that makes the denominator zero, which is x = 2. Calculate the limit of y as x approaches 2 from the left and right.

Step 5: To find horizontal asymptotes, consider the behavior of the function as x approaches infinity. Compare the degrees of the numerator and denominator. Since both have the same degree (3), the horizontal asymptote is determined by the ratio of the leading coefficients, which is 1. Therefore, y = 1 is a horizontal asymptote.

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주요 개념

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Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, like y = x³ / (x³ − 8), the domain is determined by identifying values that make the denominator zero, as these values would lead to undefined outputs.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a certain value. They are essential for analyzing the function's behavior near points of discontinuity or at infinity, which helps in determining vertical and horizontal asymptotes.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where the function is undefined (often where the denominator is zero), while horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity, indicating the function's end behavior.

Domains and AsymptotesDetermine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.y = x³ / (x³ − 8)