What is the limit of the greatest integer function?

1 Answer
Mar 22, 2016

Answer:

See explanation...

Explanation:

The "greatest integer" function otherwise known as the "floor" function has the following limits:

#lim_(x->+oo) floor(x) = +oo#

#lim_(x->-oo) floor(x) = -oo#

If #n# is any integer (positive or negative) then:

#lim_(x->n^-) floor(x) = n-1#

#lim_(x->n^+) floor(x) = n#

So the left and right limits differ at any integer and the function is discontinuous there.

If #a# is any Real number that is not an integer, then:

#lim_(x->a) floor(x) = floor(a)#

So the left and right limits agree at any other Real number and the function is continuous there.