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Chapter 5: Problem 68

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

Short Answer

Expert verified
The limit of \(\tanh x\) as \(x\) approaches infinity is \(1\).

Step by step solution

01

Identify the type of function

The given function is a Hyperbolic tangent function, denoted as \(\tanh x\).
02

Know the properties of function

\(\tanh x\) is a type of sigmoid function, which ranges between \(-1\) and \(1\). The limit of \(\tanh x\) as \(x\) approaches infinity is \(1\), and the limit of \(\tanh x\) as \(x\) approaches negative infinity is \(-1\).
03

Calculate the limit

Apply the property identified in Step 2. The limit as \(x\) approaches infinity for \(\tanh x\) is \(1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions but for a hyperbola, not a circle. They are often represented by 'sinh' for hyperbolic sine and 'cosh' for hyperbolic cosine, with 'tanh' being the hyperbolic tangent function. An intriguing attribute of these functions is their relationship with exponential functions. For example, \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\).

Hyperbolic functions appear in many areas such as calculus, physics, and engineering. They are particularly useful in solving problems involving hyperbolic angles and the geometry of hyperbolas.
Limits in Calculus
Limits are foundational in calculus, allowing mathematicians to precisely define concepts such as continuity, derivatives, and integrals. In essence, a limit captures the value that a function approaches as the input approaches a particular point. This point could be a finite number, infinity, or negative infinity. The notation \(\lim_{x \to a} f(x) = L\) is read as 'the limit of f(x) as x approaches a equals L'. This concept is vital for understanding the behavior of functions at points that are not easily evaluated directly.
Sigmoid Functions
Sigmoid functions are distinguished by their distinctive 'S'-shaped curve, often used in contexts where a function needs to map any real-valued number into a small range such as between 0 and 1, or -1 and 1. The hyperbolic tangent function, \(\tanh(x)\), is a popular example of a sigmoid function because of its useful properties in fields like artificial intelligence and statistics. One key property of sigmoid functions is that they tend to a fixed value as the input increases or decreases without bound. This makes them ideal for modeling situations where a rate of change decreases over time until it stabilizes, such as in learning processes.
Behavior of tanh Function
The hyperbolic tangent function, denoted as \(\tanh x\), exhibits specific behavior as the value of x becomes very large or very small. As x tends to infinity, the \(\tanh x\) approaches 1, while as x tends to negative infinity, \(\tanh x\) approaches -1. This asymptotic behavior is consistent with the characteristic sigmoid shape: a smooth and continuous curve that transitions from one horizontal asymptote to another. The \(\tanh x\) function is fairly simple to evaluate at these extremes, making it a stark contrast to other trigonometric and hyperbolic functions that oscillate without settling to a fixed value.

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Most popular questions from this chapter

Find \(\left(f^{-1}\right)^{\prime}(a)\) for the function \(f\) and the given real number \(a\). \(f(x)=x^{3}-\frac{4}{x}, \quad a=6\)

Find any relative extrema of the function. Use a graphing utility to confirm your result. $$ g(x)=x \operatorname{sech} x $$

Without integrating, state the integration formula you can use to integrate each of the following. (a) \(\int \frac{e^{x}}{e^{x}+1} d x\) (b) \(\int x e^{x^{2}} d x\)

Find the derivative of the function. $$ y=x \tanh ^{-1} x+\ln \sqrt{1-x^{2}} $$

Find the derivative of the function. $$ y=\tanh ^{-1}(\sin 2 x) $$
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