Problem 19 - Essential Calculus Early Transcendentals

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) \(\lim _{x \rightarrow 0^{+}}\) 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: Limit of tanh as x approaches infinity

Recall that the hyperbolic tangent function is defined as \( \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \). As \( x \) approaches infinity, the term \( e^{-x} \) becomes negligible compared to \( e^x \). Thus, \( \tanh x \approx \frac{e^x}{e^x} = 1 \). Therefore, \( \lim_{x \to \infty} \tanh x = 1 \).

Step 2: Limit of tanh as x approaches negative infinity

When \( x \) approaches negative infinity, the roles of the exponential terms switch, making \( e^x \) negligible compared to \( e^{-x} \). Therefore, \( \tanh x \approx \frac{-e^{-x}}{e^{-x}} = -1 \). Thus, \( \lim_{x \to -\infty} \tanh x = -1 \).

Step 3: Limit of sinh as x approaches infinity

The hyperbolic sine function is defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \). As \( x \) approaches infinity, \( e^{-x} \) becomes negligible, so the expression approximates to \( \frac{e^x}{2} \). Therefore, \( \lim_{x \to \infty} \sinh x = \infty \).

Step 4: Limit of sinh as x approaches negative infinity

When \( x \) approaches negative infinity, \( e^x \) becomes negligible and \( e^{-x} \) dominates, thus \( \sinh x \approx -\frac{e^{-x}}{2} \). Therefore, \( \lim_{x \to -\infty} \sinh x = -\infty \).

Step 5: Limit of sech as x approaches infinity

The hyperbolic secant function is defined as \( \operatorname{sech} x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}} \). As \( x \rightarrow \infty \), \( e^x \) dominates, and \( \cosh x \approx \frac{e^x}{2} \). Thus, \( \operatorname{sech} x \approx \frac{2}{e^x} \), which approaches 0. Hence, \( \lim_{x \to \infty} \operatorname{sech} x = 0 \).

Step 6: Limit of coth as x approaches infinity

Coth is defined by \( \coth x = \frac{\cosh x}{\sinh x} \). As \( x \rightarrow \infty \), both \( \sinh x \) and \( \cosh x \) approach \( \frac{e^x}{2} \), making \( \coth x \approx 1 \). Therefore, \( \lim_{x \to \infty} \operatorname{coth} x = 1 \).

Step 7: Limit of coth as x approaches 0 from the positive side

As \( x \rightarrow 0^+ \), \( \sinh x \approx x \) and \( \cosh x \approx 1 \). Thus, \( \coth x = \frac{\cosh x}{\sinh x} \approx \frac{1}{x} \), which approaches infinity. Therefore, \( \lim_{x \to 0^+} \operatorname{coth} x = \infty \).

Step 8: Limit of coth as x approaches 0 from the negative side

Similar to the positive side, \( \sinh x \approx x \) for \( x \approx 0 \), \( \coth x = \frac{\cosh x}{\sinh x} \approx \frac{1}{x} \). Hence, as \( x \to 0^- \), \( \frac{1}{x} \) approaches \(-\infty \). Therefore, \( \lim_{x \to 0^-} \operatorname{coth} x = -\infty \).

Step 9: Limit of csch as x approaches negative infinity

The hyperbolic cosecant function is defined as \( \operatorname{csch} x = \frac{1}{\sinh x} \). As \( x \to -\infty \), \( \sinh x \to -\infty \). Therefore, \( \operatorname{csch} x \to 0 \). Thus, \( \lim_{x \to -\infty} \operatorname{csch} x = 0 \).

Key Concepts

Limits of Hyperbolic Functions

Hyperbolic functions have interesting behavior when it comes to limits. Understanding these limits is crucial in math and physics.

• Limits at infinity or negative infinity often show asymptotic behavior.
• For instance, as x approaches positive or negative infinity, some hyperbolic function limits reach constant values, indicating a horizontal asymptote.
• Others approach infinity or negative infinity, reflecting rapid growth in magnitudes.

By examining these behaviors, you gain insights into each hyperbolic function's range and asymptotic nature. For example, the limit of sinh(x) as x approaches infinity is essentially infinity because the exponential function grows rapidly with increasing x, overwhelming any constant subtraction. Similarly, as x approaches negative infinity, the behavior flips, leading to negative infinity for sinh.
Knowledge of these limits provides a solid foundation for analyzing the intrinsic properties of hyperbolic functions.

Tanh Function

The hyperbolic tangent function, denoted as tanh, is defined by the equation \( \tanh x = \frac{\sinh x}{\cosh x} \).
Here are some key features of the tanh function:

• The tanh function bears a semblance to the tangent function in trigonometry but retains its unique hyperbolic properties.
• As x approaches infinity, \( \tanh x \) approaches 1. This occurs because the term \( e^{-x} \) becomes negligible, making sinh and cosh roughly equal, resulting in the ratio tending towards 1.
• Conversely, as x approaches negative infinity, \( \tanh x \) approaches -1. Here, \( e^x \) becomes negligible compared to \( e^{-x} \), reversing the ratio.

This behavior gives tanh a range from -1 to 1, making it particularly useful in neural networks for bounding output values.

Sinh Function

The hyperbolic sine function, represented as sinh, is defined by the equation \( \sinh x = \frac{e^x - e^{-x}}{2} \).
It's an important function in hyperbolic geometry:

• As x heads towards positive infinity, \( e^x \) escalates rapidly, largely dictating sinh's behavior. Thus, \( \lim_{x \to \infty} \sinh x = \infty \).
• For negative infinity, the role swaps; here, \( e^{-x} \) becomes dominant, and \( \sinh x \) approaches \(-\infty \).

These tendencies highlight the unbounded nature of the sinh function. Unlike the tanh function, sinh is not limited and can assume any real value. This quality makes it crucial in many areas like calculating hyperbolic angles or appearances in the description of certain physical phenomena such as the shape of a hanging cable.

Sech Function

The hyperbolic secant function, abbreviated as sech, is defined to complement the hyperbolic cosine. Its equation is \( \operatorname{sech} x = \frac{2}{e^x + e^{-x}} \).
This function provides a decreasing contribution as x increases:

• As x reaches infinity, the exponential term \( e^x \) dominates, making \( \operatorname{sech} x \approx \frac{2}{e^x} \), which approaches 0.

The sech function is distinctive because it starts at 1 and approaches 0, unlike many other hyperbolic functions that expand unboundedly.
This behavior positions it as a vital tool in assessing decay rates or modeling certain types of oscillations in physics.

Coth Function

The hyperbolic cotangent function, denoted by coth, is described using \( \coth x = \frac{\cosh x}{\sinh x} \).
It's essential for understanding asymptotic behavior:

• As x approaches large positive values, both sinh and cosh approach similar values, leading \( \coth x \) to approximately 1.
• Conversely, as x tends towards zero from the positive side, \( \coth x \) significantly rises to infinity due to \( \operatorname{coth} x \approx \frac{1}{x} \).
• Therefore, approaching zero from the negative side results in \(-\infty \), maintaining symmetry.

The coth function, therefore, diverges near zero, offering an infinite range transformation for small positive or negative x.

Csch Function

The hyperbolic cosecant function, or csch, is given by the formula \( \operatorname{csch} x = \frac{1}{\sinh x} \).
This function's purposes vary considerably from its counterparts:

• As x shifts towards negative infinity, \( \operatorname{csch} x \) approaches 0.

The csch function is defined everywhere sinh is non-zero and excludes x = 0 for practical calculations.
It's particularly useful in signal processing and other mathematical models requiring reciprocal transformations.