If I recall, his rule was to the effect that the ratio of the limits, when the limits are 0 or

, is the ratio of the derivatives. That said, you certainly cannot apply it to a regular limit.
As an example, consider the limit:
Lim x->0 (cos(x)+3) /(x+4)
It is clear that the above limit is 4/5.
On the other hand,
Lim x-> 0 (-sin(x))/(1) = 0.
Carl