Problem 44 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Explain your answer based on the left-hand and right-hand limits. Answer: No, the limit of the Heaviside function as x approaches 0 does not exist. This is because the left-hand limit as x approaches 0 (\(\lim _{x \rightarrow 0^{-}} H(x) = 0\)) is not equal to the right-hand limit as x approaches 0 (\(\lim _{x \rightarrow 0^{+}} H(x) = 1\)). Since these two limits are not equal, the overall limit does not exist.

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

The Heaviside function is defined as 0 when x < 0 and 1 when x ≥ 0. We can visualize the function on the interval [-1,1] as follows: 1. The function is 0 for all x < 0, so when -1 ≤ x < 0, it should be 0. 2. The function becomes 1 for all x ≥ 0, so when 0 ≤ x ≤ 1, it should be 1. So on the interval [-1,1], the Heaviside function will have a value of 0 on the left side and a value of 1 on the right side. It jumps discontinuously from 0 to 1 at x = 0.

Step 2: Investigate the left-hand limit as x approaches 0

The left-hand limit as x approaches 0 (notated as \(\lim _{x \rightarrow 0^{-}} H(x)\)) is the value of the function as x approaches 0 from the left side. Since on the left side we have H(x) = 0 for all x < 0, the left-hand limit of H(x) as x approaches 0 is 0, i.e.: $$\lim _{x \rightarrow 0^{-}} H(x) = 0$$

Step 3: Investigate the right-hand limit as x approaches 0

The right-hand limit as x approaches 0 (notated as \(\lim _{x \rightarrow 0^{+}} H(x)\)) is the value of the function as x approaches 0 from the right side. Since on the right side we have H(x) = 1 for all x ≥ 0, the right-hand limit of H(x) as x approaches 0 is 1, i.e.: $$\lim _{x \rightarrow 0^{+}} H(x) = 1$$

Step 4: Determine whether the limit as x approaches 0 exists

Now that we have examined both the left-hand and right-hand limits as x approaches 0, we can determine whether the overall limit exists. If the left-hand limit and the right-hand limit have the same value, then the limit exists and is equal to that value. However, if the left-hand limit and the right-hand limit do not have the same value, the overall limit does not exist. In our case, we have: $$\lim _{x \rightarrow 0^{-}} H(x) = 0$$ $$\lim _{x \rightarrow 0^{+}} H(x) = 1$$ Since these two limits are not equal, the overall limit of H(x) as x approaches 0 does not exist: $$\lim _{x \rightarrow 0} H(x) \text{ does not exist}$$

Key Concepts

Piecewise Function

A piecewise function is a type of function defined by multiple sub-functions, each applicable to a specific interval within the domain. The Heaviside function is a classic example of a piecewise function. It is defined as:

• 0 when \( x < 0 \)
• 1 when \( x \geq 0 \)

The function takes on different values depending on whether the input is less than or greater than zero. Visualizing a piecewise function often involves considering each piece separately and then "stitching" them together over the specified intervals. In the case of the Heaviside step function, the graph remains flat at zero until \( x = 0 \), where it jumps to one, remaining steady again for \( x \geq 0 \). This creates a dramatic transition in the graph, characteristic of piecewise functions with step changes.

Limits

Limits are used in mathematics to analyze the behavior of functions as they approach a particular point. For piecewise functions, limits help us understand how the function behaves from different directions. In the context of the Heaviside function, we examine both the left-hand limit and the right-hand limit as x approaches 0.

• Left-hand limit, \( \lim_{x \to 0^-} H(x) \), is the function value as \( x \) gets very close to 0 from values less than 0. For the Heaviside function, this is 0.
• Right-hand limit, \( \lim_{x \to 0^+} H(x) \), is the function value as \( x \) approaches 0 from values greater than or equal to 0. For the Heaviside function, this is 1.

When these two one-sided limits are not equal, as is the case here, the overall limit at that point does not exist. It indicates a point of discontinuity in the function.

Discontinuity

Discontinuity in a function occurs where there is a sudden change in value, breaking the smooth line of the graph. This is typically caused by a disruption or gap at a specific point. The Heaviside function, due to its definition, exhibits a discontinuity at \( x = 0 \).
Here’s why:

• On approaching 0 from the left (\( x < 0 \)), the function remains 0.
• On crossing 0 (\( x \geq 0 \)), the function suddenly jumps to 1.

This gap in values means the function is discontinuous at \( x = 0 \). Such discontinuities are identified where the left-hand limit and the right-hand limit do not match. Discrete jumps like this in the graph are often used to model real-world phenomena such as switches or signals, where immediate changes occur, making the Heaviside function particularly valuable in engineering studies.