1. Does the limit limx-->0:6[[x]] exist?

Question

2. Does the limit limx-->3[[x]] exist?

[[x]] is less or equal to x.

Answer 1

yeah, responding with no help isn't much, ummm, help.

lim x>n [[x]] for any integer n does not exist, since

lim x->n- = n-1
lim x->n+ = n

Since the limit from both sides is not the same, the general limit does not exist.