For the first one you don't need L'Hopital, notice that and so the limit should be (you should also make sure the hypothesis for L'Hopital's rule apply in each case, in this case they don't)
1)arctan(x)/[(1/x)-7] Mr F says: This is not an indeterminant form so
lim=infinity ..............................it's not valid to apply l'Hopital's Rule.
for this I got [(1)/(1+x^2)]/(-1/x^2) but it still is 0.
2) [1-cos(8x)]/[1-cos(9x)]
lim=0
Where are you stuck with 2)? Apply l'Hopital's rule twice and then take the limit. If you need more help, please show what you've done and say where you get stuck.