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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.AAE.3
Capitolo 7, Problema 7.AAE.3

Find the limits in Exercises 1–6.
3. lim(x→0⁺) (cox(√x))^(1/x)

Guida verificata passo dopo passo
1
Identify the limit expression: \(\lim_{x \to 0^+} \left( \cos(\sqrt{x}) \right)^{\frac{1}{x}}\).
Recognize that the expression is of the form \(f(x)^{g(x)}\) where both the base and the exponent depend on \(x\), and direct substitution leads to an indeterminate form.
Rewrite the limit using the exponential and natural logarithm to handle the exponent: \(\lim_{x \to 0^+} e^{\frac{1}{x} \ln(\cos(\sqrt{x}))}\).
Focus on finding the limit of the exponent: \(\lim_{x \to 0^+} \frac{\ln(\cos(\sqrt{x}))}{x}\). Use series expansions or approximations for \(\cos(\sqrt{x})\) near 0 to simplify the logarithm.
Apply the appropriate limit techniques (such as L'Hôpital's Rule if needed) to evaluate the exponent limit, then substitute back into the exponential to find the original limit.

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Concetti chiave

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Limit of a function as x approaches a point

The limit describes the value that a function approaches as the input approaches a specific point. Understanding one-sided limits, such as x approaching 0 from the right (0⁺), is crucial when the function behaves differently on either side of the point.
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Behavior of composite functions and substitution

When dealing with limits involving composite functions like cos(√x), it helps to analyze the inner function's behavior first. Substituting variables or expressions can simplify the limit evaluation, especially when the argument approaches zero.
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Exponential limits and the form 1^∞

Limits of the form (f(x))^(g(x)) where the base approaches 1 and the exponent approaches infinity often lead to an indeterminate form 1^∞. Applying logarithms and using L'Hôpital's Rule or series expansions helps to resolve such limits.
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Exponential Functions
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Domanda del libro di testo

17. Even-odd decompositions

b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that

f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

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Domanda del libro di testo

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

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Domanda del libro di testo

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

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Domanda del libro di testo

Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?

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Domanda del libro di testo

Find the limits in Exercises 1–6.

5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))

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Domanda del libro di testo

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

b. Find the centroid of the region.

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