Question

If the Laplace transform of \(f(t)\) and \(f'(t)\) exist and \(F(s)\) is the Laplace transform of \(f(t)\), then \[ \lim_{s\to\infty}sF(s) \] is

• A. \(f(0)f'(0)\)
• B. \(f(0)+f'(0)\)
• C. \(f'(0)\)
• D. \(f(0)\)

Hint:

Remember the Initial Value Theorem: \[ \boxed{\lim_{s\to\infty}sF(s)=f(0^+)} \] provided both \(f(t)\) and \(f'(t)\) possess Laplace transforms.

Answer 1

The Correct Option is D

Concept: The Initial Value Theorem of Laplace Transform states that if \(f(t)\) and \(f'(t)\) have Laplace transforms, then \[ f(0^+)=\lim_{s\to\infty}sF(s), \] where \(F(s)=\mathcal{L}\{f(t)\}\). This theorem provides the initial value of the function directly from its Laplace transform.

Step 1: Apply the Initial Value Theorem. Since \[ F(s)=\mathcal{L}\{f(t)\}, \] we have \[ \lim_{s\to\infty}sF(s)=f(0^+). \] For continuous functions, \[ f(0^+)=f(0). \]

Step 2: Identify the correct option. Hence, \[ \boxed{\lim_{s\to\infty}sF(s)=f(0).} \] Therefore, the correct answer is \[ \boxed{(D)\;f(0).} \]