What exactly is steady-state solution? In solving differential equation, one encounters steady-state solutions. My textbook says that steady-state solution is the limit of solutions of (ordinary) differential equations when $t \rightarrow \infty$. But the steady-state solution is given as $f(t)$, and this means that the solution is a function of $t$ - so what is this $t$ being in limit?

A steady-state solution of a differential equation is a solution which is constant in time $t$. For example the following differential equation:
$$x'(t)=x(t)-x^2(t)$$ with initial data $x(0)=c$, has two steady state solutions $x_1$ and $x_2$ which are the solutions corresponding to the initial data $c=0$ and $c=1$. As you can see, those are special solutions because they are constant in time:  $x_1(t)=0$ and $x_2(t)=1$ for all $t\geq 0$.  The other solutions are not constant but they all (except the one with initial data $c=0$) converge to $x_1$ as time goes to $\infty$. We say that the steady state "$1$" is asymptotically stable.
