Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
d. ln ∛9
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 7.7.82c
82. Use the definitions of the hyperbolic functions to find each of the following limits.
c. lim(x→∞) sinh x
Verified step by step guidance1
Recall the definition of the hyperbolic sine function: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).
Rewrite the limit using this definition: \(\lim_{x \to \infty} \sinh x = \lim_{x \to \infty} \frac{e^{x} - e^{-x}}{2}\).
Analyze the behavior of each term inside the limit as \(x\) approaches infinity: \(e^{x}\) grows without bound, while \(e^{-x}\) approaches zero.
Since \(e^{x}\) dominates \(e^{-x}\) for large \(x\), the expression behaves like \(\frac{e^{x}}{2}\) as \(x \to \infty\).
Conclude that the limit depends on the growth of \(e^{x}\), which increases without bound, so the limit tends toward infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Hyperbolic Sine Function
The hyperbolic sine function, sinh x, is defined as (e^x - e^(-x)) / 2. Understanding this definition is crucial because it expresses sinh x in terms of exponential functions, which simplifies the evaluation of limits involving sinh x.
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Definition of the Definite Integral
Behavior of Exponential Functions at Infinity
As x approaches infinity, e^x grows without bound, while e^(-x) approaches zero. Recognizing this behavior helps in simplifying expressions like sinh x by focusing on dominant terms when evaluating limits at infinity.
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5:46Graphs of Exponential Functions
Limit Evaluation Techniques
Evaluating limits often involves identifying dominant terms and applying limit laws. For sinh x as x approaches infinity, this means simplifying the expression using the exponential growth rates to determine the limit accurately.
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05:50One-Sided Limits
Related Practice
Textbook Question
What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?
Textbook Question
2. Express the following logarithms in terms of ln 5 and ln 7.
d. ln 1225
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. x²e^(-x)
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. c. arccos(√3/2)
Textbook Question
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2