Problem 34 - Calculus: Early Transcendentals

Question

The Heaviside function The Heaviside function is used in engineering applications to model flipping a switch. It is defined as $$H(x)=\left\\{\begin{array}{ll}0 & \text { if } x<0 \\\1 & \text { if } x \geq0\end{array}\right.$$ a. Sketch a graph of \(H\) on the interval [-1,1] b. Does \(\lim _{\vec{x} \rightarrow 0} H(x)\) exist?

Step-by-Step Solution

Answer

If not, explain why. Answer: No, the limit of the Heaviside function as x approaches 0 does not exist. This is because the limit from the left side of x = 0 is 0, while the limit from the right side of x = 0 is 1. Since these two limits are not equal, the overall limit of the Heaviside function as x approaches 0 does not exist.

Step 1: Sketch the graph of H on the interval [-1, 1]

To sketch the graph of the Heaviside function, we need to look at the definition of the function for different intervals of x. The function is defined as 0 for x < 0 and 1 for x ≥ 0. On the interval [-1,1], this means that the function will be 0 from x = -1 to x = 0 (non-inclusive) and it will be 1 from x = 0 (inclusive) to x = 1. To draw the graph, plot a horizontal line at y = 0 from x = -1 to x = 0 (non-inclusive) and a horizontal line at y = 1 from x = 0 (inclusive) to x = 1. There will be an open circle at x = 0, y = 0 to indicate that the function is not defined at that point for x < 0.

Step 2: Determine if the limit of H(x) as x approaches 0 exists

To evaluate the limit of the Heaviside function as x approaches 0, we need to consider the limit from both the left side and the right side of x = 0. If these two limits are equal, then the limit of the function as x approaches 0 exists. Otherwise, the limit does not exist. As x approaches 0 from the left side, we need to evaluate the function for x < 0. According to the definition of the Heaviside function, H(x) = 0 for x < 0. Therefore, the limit of H(x) as x approaches 0 from the left side (denoted as x → 0⁻) is 0. As x approaches 0 from the right side, we need to evaluate the function for x ≥ 0. According to the definition of the Heaviside function, H(x) = 1 for x ≥ 0. Therefore, the limit of H(x) as x approaches 0 from the right side (denoted as x → 0⁺) is 1. Since the left and right limits are not equal (0 ≠ 1), the limit of the Heaviside function as x approaches 0 does not exist: $$\lim _{x \rightarrow 0} H(x) \text{ does not exist.}$$

Key Concepts

Piecewise Functions

Piecewise functions are functions that have different expressions based on different intervals of the input variable. In simple terms, these functions can be thought of as a combination of multiple functions connected together. They are like separate pieces that make a whole.
In the case of the Heaviside function, it's an excellent example of a piecewise function. It consists of two pieces:

• The function is 0 when the input is less than 0, which is written as: if \(x < 0\), then \(H(x) = 0\).
• The function is 1 when the input is greater than or equal to 0, which is written as: if \(x \geq 0\), then \(H(x) = 1\).

This structure helps to model situations where a system behaves distinctly different in separate regions, such as the flipping of a switch, represented by an "on" and "off" state.

Limits

When we talk about limits in calculus, we're discussing what value a function approaches as the input approaches a particular point. This is crucial when trying to understand the behavior of a function at points where it's not continuous or is defined piecewise.
For the Heaviside function, finding the limit as \(x\) approaches zero helps us understand the transition from one piece of the function to another.
Considering limits from two directions:

• **From the left**: As \(x\) approaches 0 from the negative side (\(x \to 0^−\)), the function value stays at 0 since the function is defined as 0 for all \(x < 0\). Thus, \(\lim_{x \to 0^-} H(x) = 0\).
• **From the right**: As \(x\) approaches 0 from the positive side (\(x \to 0^+\)), the function jumps to 1 because \(x \geq 0\). Thus, \(\lim_{x \to 0^+} H(x) = 1\).

Because these one-sided limits are not equal (0≠1), the overall limit at \(x = 0\) does not exist, indicating a jump discontinuity.

Step Functions

Step functions are a special type of piecewise function characterized by "steps," where function values are constant over specific intervals. These functions resemble staircases and are useful in modeling abrupt changes.
The Heaviside function is a classic example of a step function. It has two constant steps:

• A step at 0, where the function value stays constant at 0 for all \(x < 0\).
• A step at 1, where the function value switches to 1 for all \(x \geq 0\).

This function can effectively represent systems that switch states instantly, such as an on-off electrical switch. Step functions are powerful in applications requiring analysis of systems at discrete changes, offering clear visual and mathematical insight into these sudden shifts.