Domanda del libro di testo
In Exercises 79–84, solve for y.
81. 9e^(2y) = = x^2
30
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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.PE.91
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
91. lim(x→π/2⁻) (sec(7x))(cos(3x))
Guida verificata passo dopo passo1
First, rewrite the limit expression to understand its behavior as \(x\) approaches \(\frac{\pi}{2}\) from the left: \(\lim_{x \to \frac{\pi}{2}^-} (\sec(7x)) (\cos(3x))\).
Recall that \(\sec(\theta) = \frac{1}{\cos(\theta)}\), so rewrite the expression as \(\lim_{x \to \frac{\pi}{2}^-} \frac{\cos(3x)}{\cos(7x)}\).
Evaluate the numerator and denominator separately at \(x = \frac{\pi}{2}\) to check if the limit is an indeterminate form: \(\cos(3 \cdot \frac{\pi}{2})\) and \(\cos(7 \cdot \frac{\pi}{2})\).
If direct substitution results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), apply l’Hôpital’s Rule by differentiating the numerator and denominator with respect to \(x\):
\[\lim_{x \to \frac{\pi}{2}^-} \frac{\frac{d}{dx} \cos(3x)}{\frac{d}{dx} \cos(7x)} = \lim_{x \to \frac{\pi}{2}^-} \frac{-3 \sin(3x)}{-7 \sin(7x)} = \lim_{x \to \frac{\pi}{2}^-} \frac{3 \sin(3x)}{7 \sin(7x)}.\]
Finally, evaluate this new limit by substituting \(x = \frac{\pi}{2}\) and simplify to find the limit.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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.
Video consigliato:
5:50
Power Rules
Trigonometric Limits and Continuity
Understanding the behavior of trigonometric functions near specific points is essential. Knowing how functions like secant and cosine behave as their arguments approach values like π/2 helps in simplifying and evaluating limits.
Video consigliato:
05:34Intro to Continuity
One-Sided Limits
One-sided limits consider the behavior of a function as the input approaches a point from only one side (left or right). This is important when the function behaves differently on either side, such as near vertical asymptotes or discontinuities.
Video consigliato:
05:50One-Sided Limits
Pratica correlata
Domanda del libro di testo
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
a. f(x) = 3^(-x), g(x) = 2^(-x)
28
views
Domanda del libro di testo
In Exercises 125–128 solve the differential equation.
127. yy' = sec(y²)sec²(x)
21
views
Domanda del libro di testo
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
104. lim(x→4) (sin²(πx))/(e^(x-4) + 3 - x)
40
views
Domanda del libro di testo
113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).
4
views
Domanda del libro di testo
109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
e. f(x) = arccsc(x), g(x) = 1/x
30
views