find the limits of the following:
1)arctan(x)/[(1/x)-7]
lim=infinity
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
For the first one you don't need L'Hopital, notice that $\displaystyle \lim_{x \rightarrow \infty } \arctan(x)= \pi /2$ and $\displaystyle \lim_{ x\rightarrow \infty } \frac{1}{x} -7 = -7$ so the limit should be $\displaystyle -\pi /14$ (you should also make sure the hypothesis for L'Hopital's rule apply in each case, in this case they don't)
For the second one use L'Hopital twice