Problem 82 - Thomas Calculus in SI Units

Question

Use the definitions of the hyperbolic functions to find each of the following limits. a. \(\lim _{x \rightarrow \infty} \tanh x\) b. \(\lim _{x \rightarrow-\infty} \tanh x\) c. \(\lim _{x \rightarrow \infty} \sinh x\) d. \(\lim _{x \rightarrow-\infty} \sinh x\) e. \(\lim _{x \rightarrow \infty} \operatorname{sech} x\) f. \(\lim _{x \rightarrow \infty} \operatorname{coth} x\) g. \(\lim _{x \rightarrow 0^{+}} \operatorname{coth} x\) h. \(\lim _{x \rightarrow 0^{-}} \operatorname{coth} x\) i. \(\lim _{x \rightarrow-\infty} \operatorname{csch} x\)

Step-by-Step Solution

Answer

a. 1, b. -1, c. ∞, d. -∞, e. 0, f. 1, g. ∞, h. -∞, i. 0.

Step 1: Understand the hyperbolic functions definitions

Hyperbolic functions are defined using exponential functions. For this exercise, we need to recall the formulas:\[ \sinh x = \frac{e^x - e^{-x}}{2} \]\[ \cosh x = \frac{e^x + e^{-x}}{2} \]\[ \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \]\[ \operatorname{sech} x = \frac{1}{\cosh x} \]\[ \operatorname{coth} x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}} \]\[ \operatorname{csch} x = \frac{1}{\sinh x} \]

Step 2: Evaluate \( \lim_{x \to \infty} \tanh x \)

According to the definition, \( \tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}} \). As \( x \to \infty \), \( e^x \to \infty \) and \( e^{-x} \to 0 \). Thus, the expression simplifies to \( \frac{e^x - 0}{e^x + 0} = 1 \). Therefore, \( \lim_{x \to \infty} \tanh x = 1 \).

Step 3: Evaluate \( \lim_{x \to -\infty} \tanh x \)

For \( \tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}} \) as \( x \to -\infty \), \( e^x \to 0 \) and \( e^{-x} \to \infty \). The expression becomes \( \frac{0 - e^{-x}}{0 + e^{-x}} = -1 \). Hence, \( \lim_{x \to -\infty} \tanh x = -1 \).

Step 4: Evaluate \( \lim_{x \to \infty} \sinh x \)

From the definition, \( \sinh x = \frac{e^x - e^{-x}}{2} \). As \( x \to \infty \), \( e^x \to \infty \) and \( e^{-x} \to 0 \), making \( \sinh x \to \infty \). Hence, \( \lim_{x \to \infty} \sinh x = \infty \).

Step 5: Evaluate \( \lim_{x \to -\infty} \sinh x \)

For \( \sinh x = \frac{e^x - e^{-x}}{2} \) as \( x \to -\infty \), \( e^x \to 0 \) and \( e^{-x} \to \infty \). Therefore, \( \sinh x \to -\frac{e^{-x}}{2} = -\infty \). So, \( \lim_{x \to -\infty} \sinh x = -\infty \).

Step 6: Evaluate \( \lim_{x \to \infty} \operatorname{sech} x \)

Since \( \operatorname{sech} x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}} \), as \( x \to \infty \), \( e^x \to \infty \) and \( e^{-x} \to 0 \). This makes \( \operatorname{sech} x = \frac{2}{\infty} = 0 \). Therefore, \( \lim_{x \to \infty} \operatorname{sech} x = 0 \).

Step 7: Evaluate \( \lim_{x \to \infty} \operatorname{coth} x \)

Coth is defined as \( \operatorname{coth} x = \frac{e^x + e^{-x}}{e^x - e^{-x}} \). As \( x \to \infty \), both \( e^x \to \infty \) and \( e^{-x} \to 0 \). The result simplifies to \( \frac{e^x + 0}{e^x - 0} = 1 \). Hence, \( \lim_{x \to \infty} \operatorname{coth} x = 1 \).

Step 8: Evaluate \( \lim_{x \to 0^{+}} \operatorname{coth} x \)

When analyzing \( \operatorname{coth} x = \frac{\cosh x}{\sinh x} \) as \( x \to 0^{+} \), \( \sinh x \to 0 \) and \( \cosh x \to 1 \). This results in a division by zero scenario from positive values, suggesting \( \lim_{x \to 0^{+}} \operatorname{coth} x = \infty \).

Step 9: Evaluate \( \lim_{x \to 0^{-}} \operatorname{coth} x \)

As \( \operatorname{coth} x = \frac{\cosh x}{\sinh x} \) and \( x \to 0^{-} \), \( \sinh x \to 0 \) and \( \cosh x \to 1 \). This leads again to a division by zero but from negative values, indicating \( \lim_{x \to 0^{-}} \operatorname{coth} x = -\infty \).

Step 10: Evaluate \( \lim_{x \to -\infty} \operatorname{csch} x \)

Since \( \operatorname{csch} x = \frac{1}{\sinh x} \) and \( \sinh x \to -\infty \) as \( x \to -\infty \), \( \operatorname{csch} x = \frac{1}{-\infty} = 0 \). Thus, \( \lim_{x \to -\infty} \operatorname{csch} x = 0 \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions may sound complex, but they are simply analogs of the regular trigonometric functions that we're familiar with in calculus. They are tied to exponential functions rather than angles and circles. The main hyperbolic functions include sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent), among others.
These functions are useful in various applications such as engineering, physics, and hyperbolic geometry. Unlike trigonometric functions that use the unit circle, hyperbolic functions involve the hyperbola. Let's break them down:

• extbf{Sinh}: Represents the hyperbolic sine.
• extbf{Cosh}: Represents the hyperbolic cosine.
• extbf{Tanh}: Represents the hyperbolic tangent.

Each of these is defined using exponential functions, making them integral to calculus and complex analysis.

Limits in Calculus

When studying calculus, limits are fundamental concepts. They help us understand the behavior of functions as they approach a particular point or infinity. For hyperbolic functions, limits are essential in determining how these functions behave as the variable approaches large positive or negative values or zero.
Calculating limits involves finding out what value a function approaches as the input gets very large, very small, or extremely close to a specific number. Here's why limits matter:

• Understanding the end behavior of functions.
• Determining continuity and differentiability.
• Simplifying complex expressions, especially those involving infinity.

The limits of hyperbolic functions introduce us to their behavior, such as finding the limit of extbf{tanh} at infinity or analyzing the behavior of extbf{sinh} or extbf{cosh} as x tends towards extreme values.

Exponential Functions

Hyperbolic functions are directly related to exponential functions, enhancing their understanding. An exponential function takes on the form \( e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Exponential functions naturally occur in many areas of mathematics, including growth processes and calculus.
For hyperbolic functions:

• extbf{Sinh} is given by \( rac{e^x - e^{-x}}{2} \).
• extbf{Cosh} equates to \( rac{e^x + e^{-x}}{2} \).
• extbf{Tanh} involves both, as \( rac{e^x - e^{-x}}{e^x + e^{-x}} \).

This tight relationship means that understanding exponential characteristics—such as rapid growth or decay—can help in understanding hyperbolic function behaviors.

Tanh Function

The hyperbolic tangent, or tanh function, is similar in its role to the tangent function in trigonometry. However, it uses hyperbolic properties. The formula for extbf{tanh} is \( rac{ ext{sinh} ext{ x}}{ ext{cosh} ext{ x}} \), simplified to \( rac{e^x - e^{-x}}{e^x + e^{-x}} \).
Key properties of the tanh function include:

• Approaches 1 as \( x \) tends to \( ext{ infinity } \).
• Goes to -1 as \( x \) moves towards \( ext{ negative infinity } \).
• Shows symmetry similar to the tangent function, but within a cracking origin.

The tanh function is essential in processes like signal processing and hyperbolic calculations.

Sinh Function

The extbf{sinh}, or hyperbolic sine function, is a counterpart to the sine function, yet it derives its character from exponential properties. This function is expressed as \( rac{e^x - e^{-x}}{2} \).
When exploring limits, it's crucial to note how extbf{sinh} behaves:

• ext{sinh(x)} grows towards positive infinity as \( x \) approaches infinity.
• ext{sinh(x)} moves towards negative infinity when \( x \) heads to negative infinity.

The operation of hyperbolic sine promotes applications in modeling exponential-like behaviors, notably in physics and engineering, especially in those dealing with energy and wave dynamics.

Cosh Function

The hyperbolic cosine, denoted as extbf{cosh}, is the hyperbolic equivalent of the cosine function. It is represented by \( rac{e^x + e^{-x}}{2} \). extbf{Cosh} possesses distinct characteristics that make it a majestic function:

• It's always positive, offering its minimum value of 1 when \( x = 0 \).
• Monotonically increases as \( x \) heads towards positive or negative infinity.
• Symmetrical about the y-axis, characteristic of even functions.

This makes extbf{cosh} useful in applications ranging from electrical engineering to the prediction of hanging cable shapes, like suspension bridges.