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

lim (x²/2 − 1/x) as

b. x→0⁻

검증된 단계별 안내

Step 1: Understand the problem. We need to find the limit of the function \( \frac{x^2}{2} - \frac{1}{x} \) as \( x \) approaches 0 from the left (denoted as \( x \to 0^- \)). This means we are considering values of \( x \) that are slightly less than 0.

Step 2: Analyze the behavior of each term in the function as \( x \to 0^- \). The term \( \frac{x^2}{2} \) approaches 0 because \( x^2 \) becomes very small as \( x \) approaches 0, and dividing by 2 does not affect the limit.

Step 3: Consider the term \( \frac{1}{x} \). As \( x \to 0^- \), \( x \) is negative and very close to 0, so \( \frac{1}{x} \) becomes very large negatively (approaches \( -\infty \)).

Step 4: Combine the behavior of both terms. The term \( \frac{x^2}{2} \) approaches 0, while \( \frac{1}{x} \) approaches \( -\infty \). Therefore, the expression \( \frac{x^2}{2} - \frac{1}{x} \) will be dominated by the \( -\frac{1}{x} \) term, which approaches \( -\infty \).

Step 5: Conclude the limit. Since the dominant term \( -\frac{1}{x} \) approaches \( -\infty \), the limit of the entire expression \( \frac{x^2}{2} - \frac{1}{x} \) as \( x \to 0^- \) is \( -\infty \).

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

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 where they may not be defined. In this case, we are interested in the limit of the function as x approaches 0 from the left (denoted as x→0⁻).

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In the given limit problem, as x approaches 0, the term 1/x becomes significant, potentially leading to an indeterminate form that requires further analysis to resolve. Understanding how to manipulate these forms is essential for finding limits.

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.lim (x²/2 − 1/x) asb. x→0⁻

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Limits

One-Sided Limits

Indeterminate Forms

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

lim (x²/2 − 1/x) as

b. x→0⁻