By the end of this section, you should be able to

1. Determine system stability by locating poles in s-plane from transfer function

2. Apply the initial value theorem to compute the initial output value from Y(s)

3. Apply the final value theorem to compute the steady-state output value from Y(s)

4. Verify whether each theorem is valid by checking pole locations and existence of limits

5. Cross-check initial and steady-state predictions using a time-domain response or simulation

This section on the Initial Value Theorem and Final Value Theorem shows how to extract time-domain insight directly from a frequency-domain equation. Instead of completing a full inverse Laplace transform every time, you will learn quick checks that tell you what happens at the start of the response and where the response settles in the long run. First, you will practice identifying whether a system is stable or unstable by examining pole locations in the complex plane and confirming when the left half-plane requirement is satisfied. Next, you will use the final value theorem to determine the steady-state value of a system output, which is especially useful for step-response predictions, model validation, and sanity checks on simulation results. You will also use the initial value theorem to determine the initial state implied by a Laplace-domain expression and to verify that your model matches the physical initial conditions you intended. The examples in this section emphasize how poles shape transient behavior and why the theorems can fail when their assumptions are violated. Code resources are included so you can reproduce the calculations and apply the theorems efficiently as you move into transfer functions, system analysis, and control topics.