Problem 33 - Single Variable Calculus

Question

The Heaviside Function A generalization of the unit step function or Heaviside function \(H\) of Example 3 is the function \(H_{c}\) defined by $$ H_{c}\left(t-t_{0}\right)=\left\\{\begin{array}{ll} 0 & \text { if } t

Step-by-Step Solution

Answer

If \(c \neq 0\), then the limit as \(t\) approaches \(t_0\) of \(H_c(t-t_0)\) does not exist, because the left limit \(\lim_{t \rightarrow t_0^-} H_c(t-t_0) = 0\) and the right limit \(\lim_{t \rightarrow t_0^+} H_c(t-t_0) = c\) are not equal.

Step 1: Define the left limit

We want to find the limit as \(t\) approaches \(t_0\) from the left, which can be written as \(\lim_{t \rightarrow t_0^-} H_c(t-t_0)\). In this case, when \(t < t_0\), the function is defined as \(H_c(t-t_0) = 0\). Therefore, the left limit is: \[\lim_{t \rightarrow t_0^-} H_c(t-t_0) = 0\]

Step 2: Define the right limit

Similarly, we want to find the limit as \(t\) approaches \(t_0\) from the right, which can be written as \(\lim_{t \rightarrow t_0^+} H_c(t-t_0)\). In this case, when \(t \geq t_0\), the function is defined as \(H_c(t-t_0) = c\). Therefore, the right limit is: \[\lim_{t \rightarrow t_0^+} H_c(t-t_0) = c\]

Step 3: Determine if the left and right limits are equal

We found that the left limit is 0, and the right limit is c. If \(c \neq 0\), then the left and right limits are not equal, which means the limit as \(t\) approaches \(t_0\) does not exist. In conclusion, if \(c \neq 0\), then \(\lim_{t \rightarrow t_0} H_c(t-t_0)\) does not exist, because the left and right limits are not equal.

Key Concepts

Unit Step Function

The unit step function, often called the Heaviside function, is a mathematical function that jumps from one value to another at a specified point. It's commonly denoted as \(H(t)\) and is defined as follows:

• \(H(t) = 0\) if \(t < 0\)
• \(H(t) = 1\) if \(t \geq 0\)

This function is used in various fields such as control systems and signal processing, especially for modeling switching behaviors or events that occur at a specific point in time.
The Heaviside function effectively "turns on" at \(t = 0\), making it instrumental in systems that need a threshold or a starting point. The generalization, denoted by \(H_c(t-t_0)\), adds a level of customization by allowing different jump values \(c\) and shift points \(t_0\). This allows for modeling more complex real-world phenomena where the response may differ in magnitude or timing.

Left Limit

The concept of the left limit pertained to the behavior of a function as it approaches a particular point from the left side on a number line. In calculus, this is represented as \(\lim_{t \rightarrow t_0^-} f(t)\).

For the Heaviside function \(H_c(t-t_0)\), as \(t\) approaches \(t_0\) from the left (\(t < t_0\)), the function value is \(0\). Therefore, the left limit is denoted as:

• \(\lim_{t \rightarrow t_0^-} H_c(t-t_0) = 0\)

Understanding this concept is pivotal as it defines how the function behaves just before reaching the point \(t_0\). When analyzing whether a particular limit exists at a point, knowing the left limit is crucial.

Right Limit

The right limit is the behavior of a function as it approaches a point from the right side on the number line. This is written in calculus as \(\lim_{t \rightarrow t_0^+} f(t)\).

For the Heaviside function \(H_c(t-t_0)\), as \(t\) approaches \(t_0\) from the right (\(t \geq t_0\)), the function equals the constant \(c\). Therefore, the right limit is expressed as:

• \(\lim_{t \rightarrow t_0^+} H_c(t-t_0) = c\)

The right limit is integral in determining how the function behaves immediately after passing the point \(t_0\). Both the left and the right limit are essential to check if a function’s limit truly exists at \(t_0\).

Limits in Calculus

Limits are foundational in calculus and provide a means to determine the behavior of a function as the input approaches a certain value. When examining limits, we particularly focus on whether the "left limit" and "right limit" at a point are equal.
For a limit to exist at any particular point \(t_0\), the following criteria must be met:

• The left limit \(\lim_{t \rightarrow t_0^-} f(t)\) must exist.
• The right limit \(\lim_{t \rightarrow t_0^+} f(t)\) must exist.
• The left and right limits must be equal: \(\lim_{t \rightarrow t_0^-} f(t) = \lim_{t \rightarrow t_0^+} f(t)\).

In the context of the Heaviside function \(H_c(t-t_0)\), if \(c eq 0\), the left limit is \(0\) and the right limit is \(c\). As a result, their inequality implies that the limit of the function at \(t_0\) does not exist. This highlights the importance of limits in determining continuity and the overall behavior of mathematical functions.