Textbook Question
Solve the initial value problems in Exercises 115–120.
117. dy/dx = 1/(x√(x² - 1)), x > 1; y(2) = π
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.73
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.
73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)
Verified step by step guidance1
Identify the limit expression: \(\lim_{x \to \infty} \frac{2^x - 3^x}{3^x + 4^x}\).
Notice that as \(x\) approaches infinity, the terms with the highest exponential base will dominate both numerator and denominator. Here, compare the bases: 2, 3, and 4.
Divide both numerator and denominator by the largest exponential term in the denominator, which is \$4^x$, to simplify the expression:
\[\frac{\frac{2^x}{4^x} - \frac{3^x}{4^x}}{\frac{3^x}{4^x} + \frac{4^x}{4^x}} = \frac{\left(\frac{2}{4}\right)^x - \left(\frac{3}{4}\right)^x}{\left(\frac{3}{4}\right)^x + 1}.\]
Analyze the behavior of each term as \(x \to \infty\): since \(\frac{2}{4} = 0.5\) and \(\frac{3}{4} = 0.75\), both bases are less than 1, so their powers tend to zero.
Conclude the limit by substituting the limits of these terms into the simplified expression and evaluating the resulting constant expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits at Infinity
Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how exponential functions behave as x approaches infinity is crucial, especially since different bases grow at different rates, affecting the limit's value.
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Cases Where Limits Do Not Exist
Dominant Term in Expressions
When evaluating limits involving sums or differences of exponential functions, the term with the largest base dominates as x approaches infinity. Identifying and factoring out this dominant term simplifies the expression and helps find the limit.
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6:36Simplifying Trig Expressions
L’Hôpital’s Rule and Its Limitations
L’Hôpital’s Rule is used to evaluate indeterminate forms like 0/0 or ∞/∞ by differentiating numerator and denominator. However, in some cases, such as this problem, repeated application leads to cycling without resolution, indicating alternative methods are needed.
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05:50One-Sided Limits
Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²
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Textbook Question
In Exercises 5–8, show that each function is a solution of the given initial value problem.
5. Differential Equation: 2y + y' = 4x + 2
Initial condition: y(-1) = e² - 2
Solution candidate: y = e^(-2x) + 2x
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ (e^(√r) / √r) dr
Textbook Question
Show that the graph of the inverse of f(x)=mx+b, where m and b are constants and m≠0, is a line with slope 1/m and y-intercept -b/m.
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(5-7x)