Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
44. lim (x → 0⁺) (csc x - cot x + cos x)
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 7.6.19
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
19. lim(x→∞)arccsc(x)
Verified step by step guidance1
Recall the definition of the function involved: \(\arccsc(x)\) is the inverse cosecant function, which gives the angle \(\theta\) such that \(\csc(\theta) = x\).
Understand the domain and range of \(\arccsc(x)\): the domain is \((-\infty, -1] \cup [1, \infty)\), and the range is \([-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}]\) excluding zero.
Analyze the behavior of \(\arccsc(x)\) as \(x\) approaches infinity: since \(\csc(\theta) = \frac{1}{\sin(\theta)}\), large values of \(x\) correspond to \(\sin(\theta)\) approaching zero from the positive side.
Determine the angle \(\theta\) whose cosecant is very large positive: this angle approaches \(0\) from the positive side, because \(\csc(\theta)\) becomes large when \(\sin(\theta)\) is close to zero but positive.
Conclude that \(\lim_{x \to \infty} \arccsc(x)\) approaches \(0\) from the positive side, so the limit is \(0^+\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Function as x Approaches Infinity
This concept involves finding the value that a function approaches as the input variable x grows without bound. Understanding limits at infinity helps determine the end behavior of functions, which is crucial for analyzing asymptotic behavior and horizontal asymptotes.
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Limits of Rational Functions: Denominator = 0
Inverse Trigonometric Functions - Arccsc(x)
Arccsc(x) is the inverse cosecant function, which returns the angle whose cosecant is x. Its domain is |x| ≥ 1, and its range is typically restricted to [-π/2, 0) ∪ (0, π/2]. Knowing the properties and domain of arccsc is essential for evaluating limits involving this function.
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Behavior of Arccsc(x) as x Approaches Infinity
As x approaches infinity, arccsc(x) approaches the angle whose cosecant is infinitely large, which corresponds to an angle approaching 0 from the positive side. Understanding this behavior allows one to evaluate the limit by relating large values of x to the corresponding angle values.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 87–96.
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Solve the initial value problems in Exercises 55–58.
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Textbook Question
Solve the initial value problems in Exercises 115–120.
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Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
41. y= x arcsin(x) + √(1-x²)