Infinity $\times\ 0=$ ? indeterminate or zero?

Question

I got the following equation for a differential solution $$Q= e^{-Rt/2L} (A \cos wt + B \sin wt)$$ where $R,L,w,A,B$ are constants. If $t$ tends to infinity, find $Q$?

$e^{-Rt/2L} = 0$, if $t$ tends to infinity
$(A \cos wt + B \sin wt) = \infty$, if $t$ tends to infinity is it?
So $Q = \infty \times 0$, that is indeterminate.

then what is the value for $Q$, if $t$ tends to infinity?

The answer was $Q =0$. How? Explain me?

Answer 1

By triangle inequality, you have

$$\forall t\in \mathbb R\;\; |Q(t)|\leq (|A|+|B|)e^{-\frac{Rt}{2L}}$$

since $|\cos(wt)|\leq 1, |\sin(wt)|\leq 1$.

and

$$\lim_{t\to+\infty}e^{-\frac{Rt}{2L}}=0$$

if $R.L>0$. thus

$$\lim_{t\to +\infty}Q(t)=0.$$