Transfer function $G(s)=\frac{1}{(s+7)(s+12)}$ and input $x(t)=(1+1/t)^t$, calculate the limit of the output $y(t)$ as $t\rightarrow\infty$

Question

So I have the following:

$G(s)=\dfrac{1}{(s+7)(s+12)}$, $ȳ(s)=\dfrac{x̄(s)}{(s+7)(s+12)}$

Since $x(t)=(1+1/t)^t$ which is $e$ as t tends towards infinity, can I simply say that $L(x(t))=x̄(s)=e/s$

The question was to calculate the limit of the output $y(t)$ as t tends towards infinity. Using the converging input converging output theorem, I have

$\lim_{t\rightarrow\infty}x(t)=x_\infty=e$

$\lim_{t\rightarrow\infty}y(t)=G(0)x_\infty=e/84$

Is this correct?

Answer 1

You can use the final value theorem which states that

\begin{equation} \lim_{t\rightarrow\infty}f(t)=\lim_{s\rightarrow0} s F(s) \end{equation}

where $F(s)$ is the Laplace transfort of $f(t)$. So \begin{equation} \lim_{s\rightarrow0} s \frac{e}{s} \dfrac{1}{(s+7)(s+12)} =\frac{e}{84} \end{equation}