/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 64 Find the limit. \(\lim _{x \ri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 64

Find the limit. \(\lim _{x \rightarrow \infty} \tanh x\)

Short Answer

Expert verified
The limit of the function \( \tanh x \) as \( x \) approaches infinity is 1.

Step by step solution

01

Understand the behavior of hyperbolic functions

The hyperbolic tangent function is defined as the ratio of the hyperbolic sine function to the hyperbolic cosine function, i.e., \( \tanh x = \frac{\sinh x}{\cosh x} \). As \( x \) becomes larger and larger, both \( \sinh x \) and \( \cosh x \) trends towards infinity. However, the increase in \( \cosh x \) is faster than \( \sinh x \).
02

Find the limit

As a result, the ratio \( \frac{\sinh x}{\cosh x} \), or \( \tanh x \), approaches 1 as \( x \) approaches infinity. Therefore, \( \lim _{x \rightarrow \infty} \tanh x = 1 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Functions
Hyperbolic functions are analogs of the trigonometric functions but are based on hyperbolas rather than circles. These functions are often used in calculus, physics, and engineering to model various phenomena. A key set of hyperbolic functions includes
  • Hyperbolic sine, denoted as \( \sinh x \), defined by the formula \( \sinh x = \frac{e^x - e^{-x}}{2} \).
  • Hyperbolic cosine, denoted as \( \cosh x \), defined by \( \cosh x = \frac{e^x + e^{-x}}{2} \).
These functions exhibit exponential growth characteristics. As a result, they are especially useful when studying exponential phenomena comparable to circular trigonometric functions.

Some properties of hyperbolic functions include:
  • They are defined for all real numbers.
  • They have similar addition formulas to trigonometric functions.
  • They adhere to an identity similar to the Pythagorean identity: \( \cosh^2 x - \sinh^2 x = 1 \).
Understanding these functions is essential for grasping their role and application in calculus and related fields.
Tanh Function
The hyperbolic tangent function, or \( \tanh x \), is particularly useful in describing the ratio of the hyperbolic sine and cosine functions. It is defined as \( \tanh x = \frac{\sinh x}{\cosh x} \). This function resembles the familiar tangent function from trigonometry, but it exists in the context of hyperbolic relations.

Key properties of the \( \tanh \) function are:
  • It is odd, meaning \( \tanh(-x) = -\tanh(x) \).
  • The range of \( \tanh x \) is between -1 and 1.
  • \( \tanh x \) is continuous and differentiable across its entire domain.
As \( x \) increases toward positive or negative infinity, the values of \( \tanh x \) approach 1 or -1, respectively. This behavior makes it crucial for analyses related to limits and asymptotic behaviors in calculus.
Asymptotic Behavior
Asymptotic behavior refers to the behavior of functions as the input values become very large or very small. In calculus, this often involves understanding how functions behave as they approach certain limits.

When analyzing \( \tanh x \), it's important to consider its asymptotic behavior as \( x \to \infty \). Here's how it works:
  • Both \( \sinh x \) and \( \cosh x \) increase exponentially as \( x \to \infty \), but \( \cosh x \) grows faster than \( \sinh x \).
  • This means the fraction \( \frac{\sinh x}{\cosh x} \) or \( \tanh x \) approaches 1.
This tendency ensures that the hyperbolic tangent function stabilizes at 1, showing an asymptotic limit like many other functions. Recognizing such behavior can help in predicting how functions behave at the boundary conditions and in simplifying complex expressions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A model for a power cable suspended between two towers is given. (a) Graph the model, (b) find the heights of the cable at the towers and at the midpoint between the towers, and (c) find the slope of the model at the point where the cable meets the right-hand tower. \(y=18+25 \cosh \frac{x}{25}, \quad-25 \leq x \leq 25\)

Find any relative extrema of the function. Use a graphing utility to confirm your result. \(f(x)=x \sinh (x-1)-\cosh (x-1)\)

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{1}^{x} \frac{t^{2}}{t^{2}+1} d t $$

Verify each rule by differentiating. Let \(a>0\). $$ \int \frac{d u}{\sqrt{a^{2}-u^{2}}}=\arcsin \frac{u}{a}+C $$

If \(a_{0}, a_{1}, \ldots, a_{n}\) are real numbers satisfying \(\frac{a_{0}}{1}+\frac{a_{1}}{2}+\cdots+\frac{a_{n}}{n+1}=0\) show that the equation \(a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}=0\) has at least one real zero.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.