# 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)

Which functions of time correspond to the following Laplace transforms? What values ​​will the time functions approach as time tends to infinity?
$F\left(s\right)=\frac{1}{{s}^{2}-1}$
I assumed it was a standard Laplace transformed and wrote $\mathrm{sinh}\left(t\right)$, which is unbounded when $\underset{t\to \mathrm{\infty }}{lim}\mathrm{sinh}\left(t\right)=\mathrm{\infty }$
But the expected answer is $f\left(t\right)=-0.5{e}^{-t}+0.5{e}^{t}$, why is that?
You can still ask an expert for help

## Want to know more about Laplace transform?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

gerasseltd9
Hyperbolic sine: the odd part of the exponential function, that is
$\mathrm{sinh}x=\frac{{e}^{x}-{e}^{-x}}{2}=\frac{{e}^{2x}-1}{2{e}^{x}}=\frac{1-{e}^{-2x}}{2{e}^{-x}}$
$\mathrm{sinh}\left(t\right)=-0.5{e}^{-t}+0.5{e}^{t}$
###### Did you like this example?
Melina Barber
It's just the definition of the Hyperbolic Sine:
$\mathrm{sinh}\left(x\right)=\frac{{e}^{x}-{e}^{-x}}{2}$
As you can see
$\underset{t\to +\mathrm{\infty }}{lim}\mathrm{sinh}\left(t\right)=\underset{t\to +\mathrm{\infty }}{lim}\frac{{e}^{t}-{e}^{-t}}{2}\to +\mathrm{\infty }$