Here we try to change the product into ratio by "putting under" the fraction part:

Note how we simplified the expression that we obtained from l'Hospital before attempting anything new. This looks quite bad, but note that the ratio of polynomials can be handled quite easily using tricks from the box "polynomials and ratios with powers". First we determine dominant powers: x⁴ in the numerator and x² in the denominator. Now we should factor these out and compare, but it will be easier if we cancel the smaller of the two dominant terms, namely x². Then we are ready to put in the infinity.

We obtained "infinity over infinity", which requires another l'Hospital, probably even more. So this way it was at best as difficult as the other solution.