Question

The floor function, ⌊x⌋, is defined as: ⌊x⌋ equals the largest integer less than or
equal to x.
Determine if
(a) limx →n ⌊x⌋ and
(b) limx→n+1/2 ⌊x⌋ exist,
when n ∈ Z. If the limit exists give it.

Answer 1

As ⌊x⌋ equals the largest integer less than or equal to x,
lim(x->n-)= n-1 from the left side of n;
lim(x->n+)= n from the right side of n.

n+1/2 is always less than n+1, thus
lim(x-> n+1/2) = n.