Domanda del libro di testo
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
41. y= x arcsin(x) + √(1-x²)
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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 7.5.55
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
55. lim (x → ∞) (ln x)^(1/x)
Guida verificata passo dopo passo1
Identify the limit expression: \(\lim_{x \to \infty} (\ln x)^{\frac{1}{x}}\).
Recognize that the expression is of the form \(f(x)^{g(x)}\) where both the base and the exponent depend on \(x\). To handle limits of this form, it is often helpful to rewrite the expression using the exponential and logarithm functions.
Rewrite the expression as \(e^{\ln\left((\ln x)^{\frac{1}{x}}\right)} = e^{\frac{1}{x} \ln(\ln x)}\).
Focus on finding the limit of the exponent: \(\lim_{x \to \infty} \frac{\ln(\ln x)}{x}\). Analyze the behavior of the numerator and denominator as \(x\) approaches infinity.
Since \(\ln(\ln x)\) grows very slowly compared to \(x\), the fraction \(\frac{\ln(\ln x)}{x}\) approaches zero. Therefore, the original limit becomes \(e^0\), which you can interpret to find the final limit.
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Limits at Infinity
Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how functions like logarithms and powers behave as x approaches infinity is crucial for evaluating such limits.
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03:07
Cases Where Limits Do Not Exist
Indeterminate Forms
Indeterminate forms occur when a limit expression does not directly yield a clear value, such as 1^∞, 0^0, or ∞^0. Recognizing these forms helps in applying appropriate techniques like logarithms or L'Hôpital's Rule to find the limit.
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Percorso guidato
6:20Circles in General Form
Logarithmic Transformation for Limits
Taking the natural logarithm of a function can simplify limits involving powers by converting exponentiation into multiplication. This transformation often makes it easier to apply limit laws and L'Hôpital's Rule to evaluate complex expressions.
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Percorso guidato
5:25Intro to Transformations
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In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
18. y = t√(ln t)
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Domanda del libro di testo
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
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For Exercises 127 and 128 find a function f satisfying each equation.
127. ∫₂ˣ √(f(t)) dt = x ln x
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Solve the initial value problems in Exercises 87 and 88.
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In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
29. y = ln(1/(x√(x+1)))
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