Maple / a logical question

Jul 2009
My lecturer gave us the following question:

\(\displaystyle f(x):=\frac{x}{x^{sin(x)}-1}\)

Using Maple, find the limit of f(x) in \(\displaystyle x=0^+\).
In order to approve Maple's answer, find the answers of the following equations:
*\(\displaystyle f(x)=-0.1\) ( I got something with \(\displaystyle 10^-4 \))
*\(\displaystyle f(x)=-0.01\) ( I got something with \(\displaystyle 10^-44 \))
* \(\displaystyle f(x)=-0.001\) ( I got something with \(\displaystyle 10^-435\) )

According to your answers, how many digits should maple work with in order to find the solution of:
\(\displaystyle f(x)=-10^{-10}\)

Now, I think I got the idea, I just want you to approve / support it: I believe that if you create a series out of the answers, you get:
\(\displaystyle a_1=-4, a_2=-44, a_3=-435, ...\)
Therefore, a_10 should be something like \(\displaystyle 435 * 10^{-7}\).

Is my 'logic' right?

Thank you for your time :)