Time function of Laplace tranform $F(s) = \frac{1}{s^2-1}$

Question

I have the following question: What are the time functions corresponding to the Laplace transforms below? What values will the time functions approach as time goes to infinity?

$$F(s) = \frac{1}{s^2-1}$$

I assumed it was a standard Laplace transformed and wrote $\sinh(t)$, which is unbounded when $\lim_{t\to \infty} \sinh(t) = \infty$

But the expected answer is $f(t) = −0.5e^{−t} + 0.5e^t$, why is that?

Answer 1

See Wikipedia for the definition of $\sinh(t)$ :$$\sinh(t) = −0.5e^{−t} + 0.5e^t$$

Answer 2

It's just the definition of the Hyperbolic Sine:

$$\sinh(x) = \dfrac{e^x - e^{-x}}{2}$$

Whence your result.

As you can see

$$\lim_{t\to +\infty} \sinh(t) = \lim_{t\to +\infty} \dfrac{e^t - e^{-t}}{2} \to +\infty$$