Domanda del libro di testo
In Exercises 1–4, solve for t.
e^(sqrt(t)) = x^2
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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 7.1.84
84. Find lim(x→∞) (√(x² + 1) - √x).
Guida verificata passo dopo passo1
Identify the limit expression: \(\lim_{x \to \infty} \left( \sqrt{x^{2} + 1} - \sqrt{x} \right)\).
Recognize that as \(x\) approaches infinity, both \(\sqrt{x^{2} + 1}\) and \(\sqrt{x}\) grow large, so direct substitution leads to an indeterminate form of type \(\infty - \infty\).
To resolve this, multiply and divide the expression by the conjugate of the difference to rationalize it: multiply by \(\frac{\sqrt{x^{2} + 1} + \sqrt{x}}{\sqrt{x^{2} + 1} + \sqrt{x}}\).
Simplify the numerator using the difference of squares formula: \(\left( \sqrt{x^{2} + 1} - \sqrt{x} \right) \left( \sqrt{x^{2} + 1} + \sqrt{x} \right) = (x^{2} + 1) - x\).
Rewrite the limit as \(\lim_{x \to \infty} \frac{(x^{2} + 1) - x}{\sqrt{x^{2} + 1} + \sqrt{x}}\) and then simplify the numerator and denominator separately to analyze the behavior as \(x\) approaches infinity.
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Limits at Infinity
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