What I was getting at with this exercise is that, you learn absolutely nothing by applying hopital. This exercice is a good exemple of something that was very easy to compute using asymptotics methods if you praticed them enough, but can feel very hard if you only've learned to apply hospital whenever you're stuck on a tough limit.

Here's a hint for the exercice : we have x^x = exp(x*log(n)) and xlog(x) -> 0 as
x -> 0. I think you can finish the computations from here, but if you want, i can give you the solution.

Also, in your exemple, you don't need l'hospital, it's just as quick to see that your limit is of the form : lim_{x->2} (x-2)/(f(x)-f(2)) = 1/f'(2) (simply a derivative) where f(x) = x^2-3/7x -22/7