y remains finite for large values of x

Hi again, just a quick question about the wording of a question.

I have the General Solution to a differential equation:

$\displaystyle y = Ae^{3x} + Be^{-2x} + \frac{1}{6} (\cos{3x} - \sin{3x})$

We are given the fact that when $\displaystyle y=1$, $\displaystyle x=0$.

From this fact I've got the equation $\displaystyle A+B = \frac{5}{6}$

The only other information we are given is that y remains finite for large values of x.

Here's how I interpreted this, can someone please just verify that this is right.

When $\displaystyle x$ is large, then $\displaystyle Be^{-2x}$ tends towards 0, we can then discount this from out equation.

When $\displaystyle x$ gets larger, $\displaystyle Ae^{3x}$ eventually goes towards infinity, but the question states that y remains finite, therefore $\displaystyle A = 0$.

The final solution I have got then is:

$\displaystyle y = \frac{5}{6}e^{-2x} + \frac{1}{6} (\cos{3x} - \sin{3x})$.

Is this correct?

Thanks