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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.5.35
Chapter 7, Problem 7.5.35

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Power Rules

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.
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Properties of Functions

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.
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Limits of Rational Functions: Denominator = 0
Related Practice
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