Can somebody please show me how one computes

$\displaystyle \lim_{x \rightarrow 0^+} \left(\frac 2x \right)^{\sin (x)}$

If we assign it to the variable $\displaystyle y$ and then take the logarithm of both sides, we should make it more easier. But after using some log laws and simplifications, I end up with

$\displaystyle \ln y = \ln (2) \cd \sin (x) - \sin (x) \cd \ln (x) \, .$

Now, because of ln(x), we need to use L'Hôspitals Rule but that isn't getting me anywhere since ln(x) keeps showing up. Can somebody help?