It is possible to do it with l'Hospital's rule. It takes 4 applications, but it does work! Do the exponential transformation, and continue simplifying with l'Hospital's rule and limits until you get:
$ \exp\left(-\frac{\left(2 \left(\lim_{x\rightarrow0} \cos\left(2 x\right)\right)\right)}{\left(\lim_{x\rightarrow0} \left(\left(4 x^2-6\right) \cos\left(2 x\right)+12 x \sin\left(2 x\right)\right)\right)}\right) $
Use a bunch of limit rules (quotient, continuity, sum, product, polynomial, in that order), in order to get: $ \exp\left(-\frac{\left(2 \cos\left(\lim_{x\rightarrow0} 2 x\right)\right)}{\left(12 \left(\lim_{x\rightarrow0} x\right) \left(\lim_{x\rightarrow0} \sin\left(2 x\right)\right)-6 \cos\left(\lim_{x\rightarrow0} 2 x\right)\right)}\right) $
Just evaluate all the limits now: $ \exp\left(\frac{1}{3}\right) $
Tedious, but certainly doable! I would recommend Fabian's solution instead, 4 applications of Hospital's, although possible, is something you want to avoid if possible.