Essentially, they are ignoring terms "small" compared with the very large j: in this case that's just about everything that doesn't involve j.
I must admit that I don't understand why they include the "+1" in "j+1" in the denominator. If j is large, then 1 is certainly small compared with it. Also, there is no reason not to cancel the js in the numerator and denominator.
If j= 10000, say, then 2j/(j(j+1))= 20000/(10000*10001)= 20000/100100= 0.00019998 while 2/(j+1)= 2/(10001)= 0.00019998- exactly the same thing and 2/j= 2/10000= 0.0002.
Your use of L'Hopital's rule IS incorrect. Your first result of
aj * 2 (j) / ( j * j) = aj * 2/j is good.