Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
a. lim(x→∞) tanh 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 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 7.5.89a
89. Use limits to find horizontal asymptotes for each function.
a. y = x tan(1/x)
Verified step by step guidance1
Recall that horizontal asymptotes are found by evaluating the limits of the function as \(x\) approaches \(\infty\) and \(-\infty\).
Set up the limit for \(x \to \infty\): \(\lim_{x \to \infty} x \tan\left(\frac{1}{x}\right)\).
To evaluate this limit, consider the substitution \(t = \frac{1}{x}\), so as \(x \to \infty\), \(t \to 0^+\), and rewrite the limit as \(\lim_{t \to 0^+} \frac{\tan(t)}{t^{-1}} = \lim_{t \to 0^+} \frac{\tan(t)}{t} \cdot t\).
Use the fact that \(\lim_{t \to 0} \frac{\tan(t)}{t} = 1\) to simplify the expression and analyze the behavior of the limit.
Repeat the process for \(x \to -\infty\) by considering \(t = \frac{1}{x} \to 0^-\) and evaluate \(\lim_{x \to -\infty} x \tan\left(\frac{1}{x}\right)\) similarly to find the horizontal asymptote on the left side.
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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 variable grows very large or very small. They help determine the end behavior of functions, which is essential for identifying horizontal asymptotes by evaluating the limit of the function as x approaches infinity or negative infinity.
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Cases Where Limits Do Not Exist
Horizontal Asymptotes
A horizontal asymptote is a horizontal line that the graph of a function approaches as x tends to infinity or negative infinity. It is found by calculating the limit of the function as x approaches ±∞. If the limit exists and is finite, that value is the horizontal asymptote.
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5:50Asymptotes of Hyperbolas
Behavior of Trigonometric Functions Near Zero
Understanding how trigonometric functions behave near zero is crucial, especially for expressions like tan(1/x) as x approaches infinity. Since 1/x approaches zero, knowing that tan(θ) ≈ θ for small θ helps simplify the limit and find the horizontal asymptote.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_2(x²)
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?
a. x² + √x
Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
a. y = e^(-x)
Textbook Question
31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.
a. How long will it take to reduce the number of cases to 1000?
Textbook Question
[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
141. a. y=arccos(cos x)