Question #750

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.

Expert's answer

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.

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.

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