Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim x/(x² − 1) as

d. x→−1⁻

검증된 단계별 안내

Step 1: Understand the problem. We need to find the limit of the function \( \frac{x}{x^2 - 1} \) as \( x \) approaches \( -1 \) from the left (denoted as \( x \to -1^- \)). This involves analyzing the behavior of the function as \( x \) gets very close to \( -1 \) from values less than \( -1 \).

Step 2: Factor the denominator. The expression \( x^2 - 1 \) can be factored using the difference of squares formula: \( x^2 - 1 = (x - 1)(x + 1) \). This will help us understand the behavior of the function near \( x = -1 \).

Step 3: Analyze the sign of the denominator near \( x = -1 \). As \( x \to -1^- \), \( x + 1 \) approaches 0 from the negative side, and \( x - 1 \) is negative. Therefore, the product \( (x - 1)(x + 1) \) is positive.

Step 4: Consider the numerator. The numerator \( x \) approaches \( -1 \) as \( x \to -1^- \). Since \( x \) is negative and the denominator is positive, the overall expression \( \frac{x}{x^2 - 1} \) will be negative.

Step 5: Determine the limit. As \( x \to -1^- \), the denominator \( (x - 1)(x + 1) \) approaches 0, causing the fraction \( \frac{x}{x^2 - 1} \) to become very large in magnitude but negative in sign. Therefore, the limit is \( -\infty \).

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주요 개념

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Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit of the function as x approaches -1 from the left.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The behavior of these functions can vary significantly based on their numerator and denominator. In this limit problem, the function x/(x² - 1) is a rational function, and understanding its structure helps in determining the limit as x approaches -1, especially since the denominator can lead to undefined behavior.

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.lim x/(x² − 1) asd. x→−1⁻

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Limits

One-Sided Limits

Rational Functions

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim x/(x² − 1) as

d. x→−1⁻