Problem 64 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \( \sinh x \).

Step 2: Find the limit

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

Key Concepts

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.