Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
19. y = (sech θ)(1-ln(sech θ))
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 7.5.44
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
44. lim (x → 0⁺) (csc x - cot x + cos x)
Verified step by step guidance1
First, rewrite the expression to understand its behavior as \(x \to 0^+\). The limit is \(\lim_{x \to 0^+} (\csc x - \cot x + \cos x)\).
Recall the definitions: \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\). Substitute these into the expression to get \(\frac{1}{\sin x} - \frac{\cos x}{\sin x} + \cos x\).
Combine the terms with common denominator \(\sin x\): \(\frac{1 - \cos x}{\sin x} + \cos x\).
Analyze the limit of \(\frac{1 - \cos x}{\sin x}\) as \(x \to 0^+\). Both numerator and denominator approach 0, so this is an indeterminate form \(\frac{0}{0}\), which allows the use of l’Hôpital’s Rule.
Apply l’Hôpital’s Rule by differentiating numerator and denominator separately: differentiate \(1 - \cos x\) to get \(\sin x\), and differentiate \(\sin x\) to get \(\cos x\). Then rewrite the limit as \(\lim_{x \to 0^+} \frac{\sin x}{\cos x} + \cos x\) and evaluate 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.
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Trigonometric Limits and Functions
Understanding the behavior of trigonometric functions such as csc(x), cot(x), and cos(x) near specific points is essential. Knowing their limits and series expansions near zero helps simplify expressions and identify indeterminate forms.
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Limit Evaluation Techniques
Besides l’Hôpital’s Rule, techniques like algebraic manipulation, recognizing standard limits, and using trigonometric identities are important. These methods help transform complex expressions into forms suitable for applying l’Hôpital’s Rule or direct substitution.
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Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
14. lim (t → 0) sin 5t / 2t
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(3te^(-t))
Textbook Question
Evaluate the integrals in Exercises 87–96.
91. ∫₁^(√2) x·2^(x²) dx
Textbook Question
For Exercises 127 and 128 find a function f satisfying each equation.
127. ∫₂ˣ √(f(t)) dt = x ln x
Textbook Question
Solve the initial value problems in Exercises 87 and 88.
88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1