Domanda del libro di testo
Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.
lim x→(−π/2)⁺ sec x
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 2, Problema 52
Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.
lim θ→0 (2 − cot θ)
Guida verificata passo dopo passo1
First, recall the definition of cotangent: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This will help us rewrite the expression in terms of sine and cosine.
Substitute \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \) in the limit expression: \( \lim_{\theta \to 0} (2 - \frac{\cos \theta}{\sin \theta}) \).
To simplify the expression, find a common denominator: \( \lim_{\theta \to 0} \left( \frac{2\sin \theta - \cos \theta}{\sin \theta} \right) \).
Evaluate the limit by considering the behavior of \( \sin \theta \) and \( \cos \theta \) as \( \theta \to 0 \). Note that \( \sin \theta \to 0 \) and \( \cos \theta \to 1 \).
Apply L'Hôpital's Rule if necessary, which is used when the limit results in an indeterminate form like \( \frac{0}{0} \). Differentiate the numerator and the denominator separately and then evaluate the limit again.
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Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions near specific points, including points of discontinuity or infinity. Evaluating limits is crucial for defining derivatives and integrals, which are core concepts in calculus.
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05:50
One-Sided Limits
Cotangent Function
The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function, defined as cot(θ) = cos(θ)/sin(θ). It is important to understand its behavior, especially near critical points like θ = 0, where it approaches infinity. This function plays a significant role in trigonometric limits and can affect the outcome of limit evaluations.
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Percorso guidato
5:37Introduction to Cotangent Graph
Indeterminate Forms
Indeterminate forms occur in limit problems when direct substitution leads to ambiguous results, such as 0/0 or ∞ - ∞. Recognizing these forms is essential for applying techniques like L'Hôpital's Rule or algebraic manipulation to resolve the limit. Understanding how to handle indeterminate forms is key to successfully finding limits in calculus.
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3:56Slope-Intercept Form
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[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?
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