Question:
If the Laplace transform of a function \( f(t) \) is given by \( \dfrac{s + 3}{(s + 1)(s + 2)}, \) then \( f(0) \) is
If the Laplace transform of a function \( f(t) \) is given by \( \dfrac{s + 3}{(s + 1)(s + 2)}, \) then \( f(0) \) is
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To find \( f(0) \) from the Laplace transform, use the initial value theorem \( f(0) = \lim_{s \to \infty} s \cdot F(s) \).
Updated On: Dec 30, 2025
- 0
- \( \frac{1}{2} \)
- 1
- \( \frac{3}{2} \)
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The Correct Option is C
Solution and Explanation
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
Final Answer: (C) 1
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