Question

Show that \(\frac{1}{2}(1+\tanh n x) \rightarrow \theta(x)\) as \(n \rightarrow \infty\).

Short answer

As n tends towards infinity, \(\frac{1}{2}(1+\tanh nx)\) behaves more like a step function, specifically the Heaviside step function which is 0 for negative x and 1 for positive x, because the transition from -1 to +1 in the hyperbolic tangent function becomes increasingly rapid, and the addition of 1 and division by 2 scale and shift the function to have a range from 0 to 1 instead of -1 to +1.

Step-by-step solution

Understand the Hyperbolic Tangent Function

Hyperbolic tangent function, \(\tanh(x)\), has an asymptotic value of -1 as x approaches -infinity and +1 as x approaches +infinity. With the factor of n in its argument, as n becomes larger, the transition from -1 to +1 becomes more rapid (i.e it approximates a step function).

Understand the transformation \(\frac{1}{2}(1+\tanh nx)\)

The addition of 1 to \(\tanh nx\) and division by 2 shifts and scales the function to range from 0 to 1 instead of -1 to +1 (i.e., \(\frac{1}{2}(1-1)=0\) and \(\frac{1}{2}(1+1)=1\)). This corresponds to the range of the Heaviside step function.

Evaluate the Limit

As n approaches infinity, the function \(\tanh nx\) transitions from -1 to +1 more rapidly around x=0. Given the transformation to \(\frac{1}{2}(1+\tanh nx)\) which makes it range from 0 to 1, this approximates the Heaviside step function - 0 for negative x and 1 for positive x.

Key concepts

Hyperbolic Tangent Function

The hyperbolic tangent function, denoted as \(\tanh(x)\), is a very important function in both mathematics and physics. Similar to the trigonometric tangent function, it shows a relationship between two sides of a right-angled hyperbolic triangle. This function, however, is related to hyperbolas, not circles. The hyperbolic tangent function is defined as:

\[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

This function smoothly transitions from -1 to 1 as the input value \(x\) increases, which makes it incredibly useful for modeling situations where a value needs to gently change from one state to another and has applications in areas such as engineering and artificial intelligence.

The uniqueness of the hyperbolic tangent function comes from its properties. As \(x\) approaches negative infinity, \(\tanh(x)\) approaches -1, and as \(x\) approaches positive infinity, it approaches 1. These are known as horizontal asymptotes. When augmented by a scaling factor, like \(n\) in our exercise, the slope of the function becomes steeper around \(x = 0\). In essence, this mimics the abrupt change of a step function and thus can be used to approximate binary decisions or switches when \(n\) is very large.

Limit of a Function

The concept of the limit of a function is foundational to the field of calculus. In a simplified form, a limit tries to describe what happens to a function as the input approaches some value. Sometimes, this value is at the edge of the function's domain or infinity itself. In mathematical terms, we express this as \(\lim_{x \to c}f(x)\) or \(\lim_{x \to \infty}f(x)\), where \(c\) represents a constant value, and \(\infty\) symbolizes infinity.

In the case we are considering, we're interested in what happens to our transformed hyperbolic tangent function \(\frac{1}{2}(1+\tanh(nx))\) as \(n\) approaches infinity. Intuitively, we can think of this limit as answering the question: 'What does the graph of our function look like if we could zoom out as far as infinity?' This understanding is crucial because it allows us to predict the behavior of functions even outside the practical bounds of measurement or observation. For our exercise, identifying the limit as \(n\) goes to infinity shows us how the \(\tanh(nx)\) function begins to resemble the Heaviside step function – a leap from approaching mathematical concepts to applying them to real-world scenarios, such as signal processing or control systems.

Asymptotic Behavior

Understanding the asymptotic behavior of functions is a key part of analyzing mathematical models, especially when variables tend towards extreme values or infinity. An asymptote is a line that a curve approaches as the independent variable of a function heads toward a particular limit, usually infinity. A perfect example is the behavior of \(\tanh(x)\) which has horizontal asymptotes at \(y = -1\) and \(y = +1\).

In our exercise, the library of functions is extended by scaling \(\tanh(x)\) with a factor \(n\), which sharpens its transition near \(x = 0\). As \(n\) grows, the function's approach to it’s limiting values becomes so abrupt it effectively mimics an on-off switch, consistent with the Heaviside step function's behavior. This sharpness illustrates the function's asymptotic behavior, particularly how quickly it apes the asymptotes of \(\theta(x)\) as \(n\) increases. Clearly visualizing this behavior can be insightful for students who are trying to grasp the transition from theory to application in scientific and engineering contexts. For example, the sharpened \(\tanh(x)\) function can represent a rapid switch in electrical circuits or sudden changes in market dynamics in economics.