Textbook Question
Evaluate the integrals in Exercises 111–114.
113. ∫₁^(1/x) (1 / t) dt,x > 0
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.19
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)
Verified step by step guidance1
Identify the limit expression: \(\lim_{\theta \to \frac{\pi}{6}} \frac{\sin \theta - \frac{1}{2}}{\theta - \frac{\pi}{6}}\).
Check if the limit is an indeterminate form by substituting \(\theta = \frac{\pi}{6}\): since \(\sin \frac{\pi}{6} = \frac{1}{2}\), the numerator becomes \(\frac{1}{2} - \frac{1}{2} = 0\) and the denominator is \(\frac{\pi}{6} - \frac{\pi}{6} = 0\), so the limit is of the form \(\frac{0}{0}\).
Since the limit is an indeterminate form \(\frac{0}{0}\), apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(\theta\).
Compute the derivative of the numerator: \(\frac{d}{d\theta} (\sin \theta - \frac{1}{2}) = \cos \theta\), and the derivative of the denominator: \(\frac{d}{d\theta} (\theta - \frac{\pi}{6}) = 1\).
Rewrite the limit using these derivatives: \(\lim_{\theta \to \frac{\pi}{6}} \frac{\cos \theta}{1}\), then evaluate this limit by substituting \(\theta = \frac{\pi}{6}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits and Indeterminate Forms
Limits describe the behavior of a function as the input approaches a particular value. When direct substitution results in an indeterminate form like 0/0, special techniques such as l’Hôpital’s rule are needed to evaluate the limit.
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One-Sided Limits
l’Hôpital’s Rule
l’Hôpital’s rule states that if a limit yields an indeterminate form 0/0 or ∞/∞, the limit of the ratio of the functions equals the limit of the ratio of their derivatives, provided this latter limit exists. It simplifies evaluating tricky limits.
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Derivative of Trigonometric Functions
Understanding the derivatives of sine and cosine functions is essential when applying l’Hôpital’s rule to trigonometric limits. For example, the derivative of sin(θ) is cos(θ), which is used to find the new limit after differentiation.
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Related Practice
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