Problem 82 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

a. 1, b. -1, c. \( \infty \), d. \(-\infty\), e. 0, f. 1, g. \( \infty \), h. undefined, i. 0

Step 1: Definition of Hyperbolic Tangent

The hyperbolic tangent function is defined as \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \). To find the limits, we analyze the behavior of \( e^x \) and \( e^{-x} \) as \( x \) approaches infinity and negative infinity.

Step 2: Limit of \( \tanh(x) \) as \( x \to \infty \)

For \( x \to \infty \), the terms \( e^x \) dominate over \( e^{-x} \), so \( \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \approx \frac{e^x}{e^x} = 1 \). Therefore, \( \lim_{x \to \infty} \tanh x = 1 \).

Step 3: Limit of \( \tanh(x) \) as \( x \to -\infty \)

For \( x \to -\infty \), the terms \( e^{-x} \) dominate, giving \( \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \approx \frac{-e^{-x}}{e^{-x}} = -1 \). Hence, \( \lim_{x \to -\infty} \tanh x = -1 \).

Step 4: Limit of \( \sinh(x) \) as \( x \to \infty \)

The hyperbolic sine function is \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). As \( x \to \infty \), \( e^x \) becomes very large, so \( \sinh(x) \approx \frac{e^x}{2} \to \infty \). Therefore, \( \lim_{x \to \infty} \sinh x = \infty \).

Step 5: Limit of \( \sinh(x) \) as \( x \to -\infty \)

As \( x \to -\infty \), the term \( e^{-x} \) becomes dominant: \( \sinh(x) = \frac{e^x - e^{-x}}{2} \approx -\frac{e^{-x}}{2} \to -\infty \). Therefore, \( \lim_{x \to -\infty} \sinh x = -\infty \).

Step 6: Limit of \( \operatorname{sech}(x) \) as \( x \to \infty \)

The hyperbolic secant is \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}} \). As \( x \to \infty \), \( e^x \) dominates, so \( \operatorname{sech}(x) \approx \frac{2}{e^x} \to 0 \). Thus, \( \lim_{x \to \infty} \operatorname{sech} x = 0 \).

Step 7: Limit of \( \operatorname{coth}(x) \) as \( x \to \infty \)

The hyperbolic cotangent is \( \operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}} \). As \( x \to \infty \), \( e^x \) dominates, and \( \operatorname{coth}(x) \approx 1 \). Therefore, \( \lim_{x \to \infty} \operatorname{coth} x = 1 \).

Step 8: Limit of \( \operatorname{coth}(x) \) as \( x \to 0^+ \)

For \( x \to 0^+ \), the behavior of \( \operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)} \) is influenced by \( \sinh(x) \approx x \) and \( \cosh(x) \approx 1 + \frac{x^2}{2} \). As \( x \to 0^+ \), \( \operatorname{coth}(x) \to \frac{1}{x} \to \infty \). Thus, \(\lim_{x \to 0^+} \operatorname{coth} x = \infty \).

Step 9: Limit of \( \operatorname{coth}(x) \) as \( x \to 0 \)

This implies finding the behavior as \( x \to 0^- \) since \( \coth(x) \to -\infty \) for negative approaching 0, combining to give undefined as \( \lim_{x \to 0} \).

Step 10: Limit of \( \operatorname{csch}(x) \) as \( x \to -\infty \)

The hyperbolic cosecant is \( \operatorname{csch}(x) = \frac{1}{\sinh(x)} \). As \( x \to -\infty \), \( \sinh(x) \to -\infty \), thus \( \operatorname{csch}(x) \approx -\frac{1}{e^{-x}/2} \to 0 \). Therefore, \( \lim_{x \to -\infty} \operatorname{csch} x = 0 \).

Key Concepts

Hyperbolic Tangent (tanh)

The hyperbolic tangent function, denoted as \( \tanh(x) \), is a fundamental hyperbolic function with diverse applications in mathematics and engineering. It is defined as the ratio of the hyperbolic sine to the hyperbolic cosine: \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \). This function is known for its characteristic S-shaped curve, which approaches different limits as the input \( x \) moves towards positive or negative infinity.

Key limit values for \( \tanh(x) \) include:

• As \( x \to \infty \), \( \tanh(x) \to 1 \). This occurs because the term \( e^x \) becomes significantly larger than \( e^{-x} \), simplifying the function to approximately \( \frac{e^x}{e^x} = 1 \).
• As \( x \to -\infty \), \( \tanh(x) \to -1 \). Here, \( e^{-x} \) dominates, leading the function towards \( -1 \).

The hyperbolic tangent function is continuous and differentiable, making it an ideal choice for activation functions in neural networks, smoothing functions in numerical analysis, and various other applications.

Hyperbolic Sine (sinh)

The hyperbolic sine function, expressed as \( \sinh(x) \), is another crucial hyperbolic function. Its formula is \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). Unlike the sine function in trigonometry, the hyperbolic sine function does not oscillate, but rather increases or decreases exponentially.

When examining the limits of \( \sinh(x) \), we observe the following behavior:

• As \( x \to \infty \), \( \sinh(x) \to \infty \). This is because \( e^x \) grows significantly larger than \( e^{-x} \), forcing \( \frac{e^x - e^{-x}}{2} \to \infty \).
• As \( x \to -\infty \), \( \sinh(x) \to -\infty \). Here, the term \( e^{-x} \) becomes the dominant term in the expression, resulting in an increasingly negative value.

The increase of \( \sinh(x) \) for large positive \( x \), and decrease for large negative \( x \), is utilized in hyperbolic geometry, physics for representing relations in special relativity, and other areas involving hyperbolic curves.

Understanding Limits

In calculus, the concept of a limit is vital for understanding the behavior of functions as they approach specific points or tend toward infinity. Limits allow us to ascertain instantaneous rates of change and continuity of functions.

For hyperbolic functions such as \( \tanh \) and \( \sinh \), evaluating limits helps in analyzing their asymptotic behavior:

• For \( \lim_{x \to \infty} \), we are interested in what value a function approaches as its input grows without bound.
• For \( \lim_{x \to -\infty} \), the concern is the value as the input decreases without bound.
• Finally, understanding \( \lim_{x \to 0} \) or \( \lim_{x \to 0^+} \) helps in evaluating how functions behave very close to zero, from either the positive or negative side.

The study of limits extends to many mathematical concepts including derivatives and integrals, forming the foundation for advanced mathematical subjects.

Hyperbolic Cotangent (coth)

The hyperbolic cotangent function, \( \coth(x) \), is defined as the reciprocal of hyperbolic tangent, \( \coth(x) = \frac{\cosh(x)}{\sinh(x)} \). It is an important function, especially when working with hyperbolic identities and solutions in differential equations.

The behavior of \( \coth(x) \) is notable near various limits:

• As \( x \to \infty \), \( \coth(x) \) approaches \( 1 \), because both the numerator and denominator are dominated by the exponential growth of \( e^x \), leading to \( \frac{e^x}{e^x} \equiv 1 \).
• When \( x \) nears zero from the positive side, \( \lim_{x \to 0^+} \coth(x) \to \infty \). This is due to \( \sinh(x) \approx x \) and \( \coth(x) \approx \frac{1}{x} \), which tends to grow exponentially large as \( x \to 0^+ \).

The hyperbolic cotangent is undefined at zero due to a potential division by zero, emphasizing the need for careful limit analysis when approaching points of discontinuity.

Hyperbolic Secant (sech)

The hyperbolic secant, \( \operatorname{sech}(x) \), is another important hyperbolic function. It is defined as the reciprocal of the hyperbolic cosine function, \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}} \). This function has interesting properties when examined under limiting conditions.

Some notable limits of \( \operatorname{sech}(x) \) include:

• As \( x \to \infty \), \( \operatorname{sech}(x) \to 0 \). The term \( e^x \) in the denominator grows very large, making \( \frac{2}{e^x} \) approach zero.

Such limit behaviors make \( \operatorname{sech} \) a valuable function in fields like signal processing, where it can describe damping factors. The hyperbolic secant is also used in solving differential equations and in modeling phenomena that require characterization of rapid changes.