# What is the limit of the greatest integer function?

Mar 22, 2016

See explanation...

#### Explanation:

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

${\lim}_{x \to + \infty} \left\lfloor x \right\rfloor = + \infty$

${\lim}_{x \to - \infty} \left\lfloor x \right\rfloor = - \infty$

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

${\lim}_{x \to {n}^{-}} \left\lfloor x \right\rfloor = n - 1$

${\lim}_{x \to {n}^{+}} \left\lfloor x \right\rfloor = n$

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:

${\lim}_{x \to a} \left\lfloor x \right\rfloor = \left\lfloor a \right\rfloor$

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