limit of absolute value vs without absolute value. So suppose I know
$\lim_{t \rightarrow\infty} |u(t)| = \dfrac{|B|}{c}$, where $c$ and $B$ are real numbers.
Can I conclude that 
$$\lim_{t \rightarrow\infty} u(t) = \frac{B}{c}$$
That is, can I just drop the absolute value signs? I can't justify that I can do this for some reason...
 A: No, we cannot simply drop the absolute values.  For example, suppose that
$$ u(t) = \begin{cases} 1 & \text{$t \in [2n,2n+1),\ n\in\mathbb{N}$, and} \\
-1 & \text{$t \in [2n-1,2n),\ n\in\mathbb{N}$.}
\end{cases}$$
Then $|u(t)| = 1$ for all $t$, and so
$$ \lim_{t\to \infty} |u(t)| = 1.$$
On the other hand,
$ \lim_{t\to\infty} u(t) $
does not exist.

Less pathologically, suppose that we require $u$ to be continuous.  This still isn't good enough!  For example, suppose that $$u(t) = \frac{t}{1-t},$$ which is continuous on $(1,\infty)$.  Observe that
$$ \lim_{t\to\infty} |u(t)| = 1, 
\qquad\text{but}\qquad
\lim_{t\to \infty} u(t) = -1.$$
Here, the best we can do is say that
$$ \lim_{t\to\infty} |u(t)| = |B| \implies \lim_{t\to\infty} u(t) = \pm B,$$
which doesn't actually give us any more (or less) information.
A: You can't do that because the function may alternate between something approaching $+B/c$ and something approaching $-B/c.$ In that case the limit of the function would not exist but the limit of its absolute value would. However, in some cases there may be additional information that tells you the function is postive if $t$ is large, that would justify dropping the absolute value sign in the particular case.
Also, if $B$ happens to be $0$ then you could drop the absolute value sign because in that case, $+B/c$ is the same as $-B/c.$
