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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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