Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (x⁴ − 1)/(x^2−1)

Verified step by step guidance

Step 1: Identify the degrees of the numerator and the denominator.

Step 2: Compare the degrees of the numerator and the denominator.

Step 3: Analyze lim x→∞ f(x) by dividing each term by the highest power of x in the denominator.

Step 4: Analyze lim x→−∞ f(x) using the same method as in Step 3.

Step 5: Determine the horizontal asymptotes based on the limits found in Steps 3 and 4.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity examine the behavior of a function as the input approaches positive or negative infinity. This analysis helps determine the end behavior of the function, which is crucial for identifying horizontal asymptotes. For rational functions, the degrees of the numerator and denominator play a significant role in determining the limit.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as the input values become very large or very small. They indicate the value that the function approaches at infinity. For rational functions, horizontal asymptotes can be found by comparing the degrees of the numerator and denominator, leading to specific rules for their existence.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The behavior of these functions, especially at infinity, is influenced by the degrees of the polynomials involved. Understanding how to simplify and analyze these functions is essential for determining limits and asymptotic behavior.

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