Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
23. y = (x²+1)sech(ln x)
(Hint: Before differentiating, express in terms of exponentials and simplify.)
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 7.5.47
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
47. lim (t → ∞) (e^t + t²) / (e^t - t)
Verified step by step guidance1
Identify the limit expression: \(\lim_{t \to \infty} \frac{e^{t} + t^{2}}{e^{t} - t}\).
Check the form of the limit by analyzing the behavior of numerator and denominator as \(t \to \infty\). Both numerator and denominator tend to infinity, so the limit is of the form \(\frac{\infty}{\infty}\), which is an indeterminate form suitable for l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(t\): differentiate numerator \(\frac{d}{dt}(e^{t} + t^{2}) = e^{t} + 2t\), and denominator \(\frac{d}{dt}(e^{t} - t) = e^{t} - 1\).
Rewrite the limit using the derivatives: \(\lim_{t \to \infty} \frac{e^{t} + 2t}{e^{t} - 1}\).
Evaluate the new limit by considering the dominant terms as \(t \to \infty\) and determine the behavior of the fraction to find 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 used to evaluate 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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Guided course
5:50
Power Rules
Limits at Infinity
Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how exponential and polynomial functions behave as the variable approaches infinity is crucial for comparing growth rates and determining the limit.
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03:07Cases Where Limits Do Not Exist
Growth Rates of Exponential vs Polynomial Functions
Exponential functions grow faster than any polynomial function as the variable approaches infinity. Recognizing this helps simplify limits by identifying dominant terms, which is essential when applying l’Hôpital’s Rule or comparing terms in the numerator and denominator.
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09:29Exponential Growth & Decay
Related Practice
Textbook Question
77. The region in the first quadrant bounded by the coordinate axes, the line y=3, and the curve x=2/√(y+1) is revolved about the y-axis to generate a solid. Find the volume of the solid.
Textbook Question
Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
1. sinh x = -3/4
Textbook Question
In Exercises 5–8, show that each function is a solution of the given initial value problem.
7. Differential Equation: xy' + y = -sin(x), x>0
Initial condition: y(π/2) = 0
Solution candidate: y = cos(x)/x
Textbook Question
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
75. lim (x → ∞) e^(x²) / (x e^x)
1
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Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
115. y = (x + 1)ˣ