Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as

a. x → 0⁺

검증된 단계별 안내

Step 1: Understand the problem. We need to find the limit of the expression \( \frac{1}{x^{2/3}} + \frac{2}{(x - 1)^{2/3}} \) as \( x \to 0^+ \). This means we are approaching 0 from the positive side.

Step 2: Analyze the behavior of each term separately as \( x \to 0^+ \). For the term \( \frac{1}{x^{2/3}} \), as \( x \) approaches 0 from the positive side, \( x^{2/3} \) approaches 0, making \( \frac{1}{x^{2/3}} \) approach infinity.

Step 3: Consider the second term \( \frac{2}{(x - 1)^{2/3}} \). As \( x \to 0^+ \), \( x - 1 \) approaches -1. Therefore, \( (x - 1)^{2/3} \) approaches \((-1)^{2/3}\), which is a real number. Thus, \( \frac{2}{(x - 1)^{2/3}} \) approaches a finite value.

Step 4: Combine the behavior of both terms. The first term \( \frac{1}{x^{2/3}} \) dominates the behavior of the limit because it approaches infinity, while the second term approaches a finite value.

Step 5: Conclude the limit. Since the first term approaches infinity and dominates the expression, the limit of the entire expression as \( x \to 0^+ \) is infinity.

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주요 개념

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Limits

A limit in calculus describes the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for analyzing the behavior of functions at points where they may not be explicitly defined, such as points of discontinuity or infinity.

Indeterminate Forms

Indeterminate forms occur when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not have a clear value. Recognizing these forms is essential for applying techniques like L'Hôpital's Rule or algebraic manipulation to find the actual limit.

Behavior Near Zero

Analyzing the behavior of functions as x approaches zero involves understanding how terms in the function behave, especially when they involve powers or roots. This is important for determining whether the function approaches a finite value, infinity, or negative infinity as x approaches zero from the positive side.

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) asa. x → 0⁺