Question

If the Laplace transform of a function \( f(t) \) is given by \( \dfrac{s + 3}{(s + 1)(s + 2)}, \) then \( f(0) \) is 
 

• A. 0
• B. \( \frac{1}{2} \)
• C. 1
• D. \( \frac{3}{2} \)

Hint:

To find \( f(0) \) from the Laplace transform, use the initial value theorem \( f(0) = \lim_{s \to \infty} s \cdot F(s) \).

Answer 1

The Correct Option is C

The Laplace transform of a function \( f(t) \) is given by: \[ F(s) = \frac{s + 3}{(s + 1)(s + 2)}. \] We are asked to find \( f(0) \). Recall that the value of \( f(0) \) is given by the initial value theorem in Laplace transforms, which states: \[ f(0) = \lim_{s \to \infty} s F(s). \] Substitute the given \( F(s) \) into this formula: \[ f(0) = \lim_{s \to \infty} s \cdot \frac{s + 3}{(s + 1)(s + 2)}. \] As \( s \to \infty \), the terms \( +1 \) and \( +2 \) in the denominator become negligible, so the expression simplifies to: \[ f(0) = \lim_{s \to \infty} s \cdot \frac{s + 3}{s^2} = \lim_{s \to \infty} \frac{s + 3}{s} = 1. \] Thus, the value of \( f(0) \) is 1, corresponding to Option (C).
Final Answer: (C) 1