15.5.2. Example 2: Non-Converging Function (Invalid Case)

Let f(t)=sin(t)

In this example, we consider a function, f(t), defined as sin(t). It is important to note that the sine function oscillates indefinitely as time progresses, which means it does not settle down to a single value over time. Thus, unlike other functions that may stabilize at a certain output, sin(t) will keep varying between -1 and 1 without approaching any single limit.

Imagine the motion of a swing. If the swing keeps going back and forth indefinitely, it never really comes to rest. Just like the swing's motion is continuous and oscillatory, the sine function keeps oscillating between its maximum and minimum values.

1 F(s)= s2 +1

The second step is to find the Laplace transform of the function. For f(t) = sin(t), the Laplace transform is given by F(s) = 1/(s² + 1). This transform converts the time-domain function into the s-domain, allowing for easier analysis. The form 1/(s² + 1) indicates how this function behaves in the s-domain.

Think of the Laplace transform as translating a song into different musical notes. The original song (the sine function) is played in a way that many might not understand, but translating it into notes (the transform) can help musicians analyze and work with it more effectively.

sF(s)= s2 +1

In this step, we multiply the previously found F(s) by s to get sF(s), which equals (s² + 1). This multiplication is an essential part of applying the Final Value Theorem, as it prepares the function for the next computation phase, which involves taking a specific limit as s approaches zero.

This can be likened to preparing dough for baking. Just as adding flour to the mix helps in shaping the dough, here we are preparing our function for analysis through multiplication by s.

lim s lim sF(s)=s→0 =0 s→0 s2 +1

The next critical step involves taking the limit of sF(s) as s approaches zero. This limit evaluates to zero as shown in the equation lim (s² + 1) as s goes to zero results in 0. This represents the theoretical final value of the function according to the FVT. However, we must be careful because the results from the theorem might not hold true if the function has oscillatory behavior.

Think of observing a car's movement approaching a traffic light. You may expect it to stop at the light, but if the car is still accelerating at the red light instead of stopping, the expectation fails. Similarly, while the math suggests a result, the actual behavior of the sine function contradicts the limit.

But in reality, lim ¿ t→∞sin(t)¿ does not exist. ❌ Hence, FVT fails here due to oscillatory behavior.

Finally, we conclude that, in reality, the limit of sin(t) as t approaches infinity does not exist because it continues to oscillate between -1 and 1 indefinitely. This oscillatory nature means that the Final Value Theorem cannot be applied here, demonstrating how important it is to evaluate the behavior of the function before applying any theorems.

Consider a person standing between two very tall buildings shouting. Their voice carries back and forth between the buildings indefinitely without necessarily 'settling' or fading away. This is similar to the oscillations of the sine function which make its final value undefined.

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