Sketching Transfer function in time domain

[gfd43tg]

Homework Statement

$$ y(s) = \frac {s}{(s+1)(2s+1)} u(s) $$
Where $u(s)$ is the step function $\frac {1}{s}$

Find the output at t=0 and t= infinity

Homework Equations

The Attempt at a Solution

My question is kind of basic, so I know the final and initial value theorem

$$ \lim_{s \to 0} sY(s) = \lim_{t \to \infty} y(t) $$

But should I include the step function, or leave it out. Meaning, should I evaluate

$$ \lim_{s \to 0} \frac {s^{2}}{(s+1)(2s+1)} \frac {1}{s}$$
or rather,
$$ \lim_{s \to 0} \frac {s^{2}}{(s+1)(2s+1)} $$

The reason I am hesitating on this is because in the textbook example problem, they do not mention what the input function is, and proceed to solve without the step function. Then I solved a homework problem where they asked to match the transfer function output with a step function input, and at that time I did not even realize, so I was only doing limits of Y(s), not sY(s), and got them all right. So now my head is all jumbled up and I just want to get this thing cleared up!

 

[rude man]

You need to include U(s) = 1/s.
If you used the initial & final value theorems without the U(s) input you'd be solving the problem with the input = kδ(t), the delta function aka impulse input, where k = 1 Volt-sec. If you're not familiar with the delta function, be careful with it. Unlike u(t) which is dimensionless, δ(t) has dimension 1/t.

 

[gfd43tg]

Makes sense why I got them all right then, the s and 1/s cancel anyways