Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
124. x^(sin y) = ln y
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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
Verified step by step guidance1
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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06:21Properties 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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06:11Limits of Rational Functions: Denominator = 0
Related Practice
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(cost+lnt)
Textbook Question
132. What is special about the functions
f(x) = arcsin((1/√(x²+1)) and g(x)=arctan(1/x)?
Explain.
Textbook Question
Suppose that the range of g lies in the domain of f so that the composition fog is defined. If f and g are one-to-one, can anything be said about fog? Give reasons for your answer.
Textbook Question
135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).
Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=(x+1)², x≥-1