Without using l'Hopital's rule, find lim (x->0)
(sin (sin x))/x
You learned that lim (x->0) (sin x)/x = 0. That's all you need.
Write the limit as
lim (x->0) [ (sin (sin x))/(sin x) ] * [ (sin x)/x ]
= lim (x->0) (sin (sin x))/(sin x) * lim (x->0) (sin x)/x
Now let u = sin x. The important thing is that as x->0, u->0. So
the above limit is equal to
lim (u->0) (sin u)/u * lim (x->0) (sin x)/x
and we know that both limits are 1 so the product is 1.