Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
85. y = log₂(8t^(ln 2))
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 7.5.22
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
22. lim (x → 1) (x - 1) / (ln x - sin πx)
Verified step by step guidance1
First, verify that the limit is an indeterminate form of type \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) by substituting \( x = 1 \) into the expression \( \frac{x - 1}{\ln x - \sin \pi x} \).
Since direct substitution gives \( \frac{0}{0} \), apply l'Hôpital's Rule, which states that \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \) if the original limit is an indeterminate form.
Find the derivative of the numerator: \( f(x) = x - 1 \), so \( f'(x) = 1 \).
Find the derivative of the denominator: \( g(x) = \ln x - \sin \pi x \). Use the derivatives \( \frac{d}{dx} \ln x = \frac{1}{x} \) and \( \frac{d}{dx} \sin \pi x = \pi \cos \pi x \), so \( g'(x) = \frac{1}{x} - \pi \cos \pi x \).
Evaluate the new limit \( \lim_{x \to 1} \frac{f'(x)}{g'(x)} = \lim_{x \to 1} \frac{1}{\frac{1}{x} - \pi \cos \pi x} \) by substituting \( x = 1 \).
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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.
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Limits Involving Logarithmic and Trigonometric Functions
Understanding how logarithmic functions (like ln x) and trigonometric functions (like sin πx) behave near specific points is crucial. This helps in simplifying expressions and determining if direct substitution leads to indeterminate forms.
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Derivative Computation
Calculating derivatives of functions such as (x - 1), ln x, and sin πx accurately is essential when applying l’Hôpital’s Rule. Knowing the derivative rules for polynomials, logarithms, and trigonometric functions ensures correct application of the rule.
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Related Practice
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)
Textbook Question
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))
Textbook Question
Evaluate the integrals in Exercises 77–90.
77. ∫dx/√(-x²+4x-3)
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)