I think I have an idea what you mean. Let's take Now, we don't know the signs of and so we'll have to do different cases depending on which one we have. We may be able to do something clever with the signum function. I'll do the first component for to show you what's going on. We have
The trig functions are not going to either blow up or go to zero. Suppose Then the exponential blows up. Since we can't have that, the only option is if Otherwise, the component blows up. If, on the other hand, then it doesn't matter what and are, the component will go to zero.
Conversely, let's say we are looking at The situation will be precisely the opposite of the previous case. means a blow-up unless and means the component goes to zero regardless of what and are.
Does that help?