Textbook Question
L’Hôpital’s Rule
Find the limits in Exercises 103–110.
108. lim(x→∞)(e^x arctan(e^x))/(e^(2x)+x)
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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 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
Chapter 7, Problem 7.5.82
82. For what values of a and b is
lim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?
Verified step by step guidance1
Rewrite the limit expression clearly: \(\lim_{x \to 0} \left( \tan\left( \frac{2x}{x^3} \right) + \frac{a}{x^2} + \frac{\sin(bx)}{x} \right) = 0\).
Simplify the argument inside the tangent function: \(\frac{2x}{x^3} = \frac{2}{x^2}\). So the expression becomes \(\tan\left( \frac{2}{x^2} \right) + \frac{a}{x^2} + \frac{\sin(bx)}{x}\).
Analyze the behavior of each term as \(x \to 0\). Notice that \(\tan\left( \frac{2}{x^2} \right)\) oscillates wildly and does not have a limit, so for the overall limit to exist and be zero, the terms involving \(a\) and \(b\) must cancel or control the behavior.
Consider using series expansions or asymptotic behavior for \(\sin(bx)\) near zero: \(\sin(bx) \approx bx\) for small \(x\), so \(\frac{\sin(bx)}{x} \approx b\). This term tends to \(b\) as \(x \to 0\).
Since \(\tan\left( \frac{2}{x^2} \right)\) does not have a limit, the only way for the entire limit to be zero is if the expression inside the limit is defined differently or if the problem is reconsidered. Re-examine the problem statement or check for possible typos in the expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Function as x Approaches a Point
The limit describes the value a function approaches as the input approaches a specific point. Understanding how to evaluate limits, especially when direct substitution leads to indeterminate forms, is essential for analyzing behavior near that point.
Recommended video:
06:11
Limits of Rational Functions: Denominator = 0
Series Expansion (Taylor or Maclaurin Series)
Series expansions approximate functions near a point using polynomials. For small x, functions like tan(2x), sin(bx), and powers of x can be expanded to simplify complex expressions and identify dominant terms affecting the limit.
Recommended video:
08:26Convergence of Taylor & Maclaurin Series
Handling Indeterminate Forms and Coefficient Matching
When limits yield expressions like 0/0 or ∞ - ∞, rewriting terms via expansions helps resolve indeterminacies. Matching coefficients of powers of x ensures the limit exists and equals the desired value, allowing determination of unknown parameters a and b.
Recommended video:
6:04Introduction to Polynomial Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 77–90.
90. ∫dx/((x-2)√(x²-4x+3))
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Textbook Question
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
6. sinh(2ln x)
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
65. y = (cos θ)^(√2)
Textbook Question
44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be
b. 2 hours from now?
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ∫(from 0 to lnx) sin(e^t) dt