Which functions of time correspond to the following Laplace transforms? What values ​​will the time functions approach as time tends to infinity? F(s)=(1)/(s^2−1)

amhailim 2022-09-26 Answered
Which functions of time correspond to the following Laplace transforms? What values ​​will the time functions approach as time tends to infinity?
F ( s ) = 1 s 2 1
I assumed it was a standard Laplace transformed and wrote sinh ( t ), which is unbounded when lim t sinh ( t ) =
But the expected answer is f ( t ) = 0.5 e t + 0.5 e t , why is that?
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Answers (2)

gerasseltd9
Answered 2022-09-27 Author has 8 answers
Hyperbolic sine: the odd part of the exponential function, that is
sinh x = e x e x 2 = e 2 x 1 2 e x = 1 e 2 x 2 e x
sinh ( t ) = 0.5 e t + 0.5 e t
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Melina Barber
Answered 2022-09-28 Author has 3 answers
It's just the definition of the Hyperbolic Sine:
sinh ( x ) = e x e x 2
Whence your result.
As you can see
lim t + sinh ( t ) = lim t + e t e t 2 +
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