Domanda del libro di testo
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x² − 2x, x ≤ 1
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Ch. 7 - Transcendental Functions
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 7, Problema 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)
Guida verificata passo dopo passo1
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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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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Power Rules
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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Percorso guidato
6:04Introduction to Trigonometric Functions
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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05:50One-Sided Limits
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Domanda del libro di testo
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