find the limit of this Laplace transform lim_(epsilon -> 0) L(f_(epsilon)(x)).

Bodonimhk 2022-10-12 Answered
Let f ϵ ( x ) be defined as
f ϵ ( x ) = { 1 / ϵ , 0 x ϵ 0 , x > ϵ
I calculated the laplace transform of f ϵ ( x ) to be
L ( f ϵ ( x ) ) = 1 e p ϵ p ϵ
I want to find the limit of this Laplace transform lim ϵ 0 L ( f ϵ ( x ) )
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Answers (2)

cesantedz
Answered 2022-10-13 Author has 12 answers
You can do a Taylor expansion of the exponential e p ϵ = 1 p ϵ + ( p ϵ ) 2 / 2 + . . . The 1 will cancel, and what you are left with is
lim ϵ 0 p ϵ + ( p ϵ ) 2 / 2 + . . . p ϵ = 1
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Oscar Burton
Answered 2022-10-14 Author has 3 answers
Apply L'Hospital rule
lim ϵ 0 1 e p ϵ p ϵ = lim ϵ 0 e p ϵ = 1
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