Use l’Hôpital’s rule to find the limits in Exercises 7–52.
35. lim (x → 0⁺) ln(x² + 2x) / ln x

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First, identify the form of the limit as \( x \to 0^+ \). Substitute \( x = 0^+ \) into the expression \( \frac{\ln(x^2 + 2x)}{\ln x} \) to check if it results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

Since \( \ln x \) approaches \( -\infty \) as \( x \to 0^+ \), and \( \ln(x^2 + 2x) \) also approaches \( -\infty \) because \( x^2 + 2x \to 0^+ \), the limit is of the form \( \frac{-\infty}{-\infty} \), which is an indeterminate form suitable for l'Hôpital's Rule.

Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \( x \). The derivative of the numerator is \( \frac{d}{dx} \ln(x^2 + 2x) = \frac{2x + 2}{x^2 + 2x} \). The derivative of the denominator is \( \frac{d}{dx} \ln x = \frac{1}{x} \).

Rewrite the limit using these derivatives: \[ \lim_{x \to 0^+} \frac{\frac{2x + 2}{x^2 + 2x}}{\frac{1}{x}} = \lim_{x \to 0^+} \frac{2x + 2}{x^2 + 2x} \times x \]. Simplify the expression inside the limit before evaluating.

After simplification, evaluate the limit as \( x \to 0^+ \) by substituting \( x = 0^+ \) into the simplified expression to find the limit.

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주요 개념

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

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met. This rule simplifies complex limit problems involving logarithms or other functions.

Properties of Logarithmic Functions

Logarithmic functions, such as ln(x), have specific properties including their domain (x > 0) and behavior near zero and infinity. Understanding how ln(x) behaves as x approaches 0 from the right is crucial, as ln(x) tends to negative infinity, which affects the limit evaluation.

Limit of Functions as x Approaches 0⁺

Evaluating limits as x approaches 0 from the positive side requires careful consideration of function behavior near zero. For example, expressions inside logarithms must remain positive, and the direction of approach affects the limit's existence and value. This concept ensures the limit is taken within the function's domain.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.35. lim (x → 0⁺) ln(x² + 2x) / ln x