Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.5.47

Capitolo 7, Problema 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)

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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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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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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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Cases 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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Exponential Growth & Decay

Pratica correlata

Domanda del libro di testo

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.)

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Domanda del libro di testo

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.

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Domanda del libro di testo

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

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Domanda del libro di testo

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

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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.