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Magazine Chapter 3: Problem 19
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) \(\lim _{x \rightarrow-\infty} \operatorname{csch} x\)
Short Answer
Expert verified
(a) 1, (b) -1, (c) 鈭�, (d) -鈭�, (e) 0, (f) 1, (g) 鈭�, (h) -鈭�, (i) 0.
Step by step solution
01
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, \( \lim_{x \to \infty} \tanh x = 1 \).
02
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, \( \lim_{x \to -\infty} \tanh x = -1 \).
03
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, \( \lim_{x \to \infty} \sinh x = \infty \).
04
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, \( \lim_{x \to -\infty} \sinh x = -\infty \).
05
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, \( \lim_{x \to \infty} \operatorname{sech} x = 0 \).
06
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, \( \lim_{x \to \infty} \operatorname{coth} x = 1 \).
07
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, \( \lim_{x \to 0^+} \operatorname{coth} x = \infty \).
08
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, \( \lim_{x \to 0^-} \operatorname{coth} x = -\infty \).
09
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, \( \lim_{x \to -\infty} \operatorname{csch} x = 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limits of Hyperbolic Functions
Hyperbolic functions have interesting behavior when it comes to limits. Understanding these limits is crucial in math and physics.
Knowledge of these limits provides a solid foundation for analyzing the intrinsic properties of hyperbolic functions.
- 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.
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:
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.
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:
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 \(-\infty \).
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:
This behavior positions it as a vital tool in assessing decay rates or modeling certain types of oscillations in physics.
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.
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:
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.
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:
It's particularly useful in signal processing and other mathematical models requiring reciprocal transformations.
This function's purposes vary considerably from its counterparts:
- As x shifts towards negative infinity, \( \operatorname{csch} x \) approaches 0.
It's particularly useful in signal processing and other mathematical models requiring reciprocal transformations.
