Greatest Integer Function

The greatest integer function is also known as floor function or step function. It is written as f(x) = $[\![x ]\!]$
The value of $[\![x ]\!]$ is the largest integer that is less than or equal to x.
Example 1 : greatest integer of 2.7 = $[\![2.7 ]\!]$ So the largest integer is 2
So, $[\![2.7 ]\!]$ = 2
Example 2 : greatest integer of -1.6 = $[\![-1.6 ]\!]$ So the largest integer is -2
So, $[\![-1.6 ]\!]$ = -2
Note : When the greatest integer is positive then its largest integer will be its previous whole number and when the greatest integer is negative then its largest integer will be next negative number. If there is no decimal then the largest integer will be the number itself.

Limit of Greatest Integer Function

1) Find the limit of greatest integer function f(x) = $[\![x ]\!]$ as x approaches to 0 from left and right side.
Solution : First we will graph the given function f(x) = $[\![x ]\!]$ For the limit of largest integer from the left side we will consider red circle from the above diagram.
$\lim_{x->0^{-}}[\![x ]\!]$ = -1

For the limit of largest integer from the right side we will consider blue circle from the above diagram.
$\lim_{x->0^{+}}[\![x ]\!]$ = 0

The greatest integer function is discontinuous function as its left side and right side limit gives us different values.

2) Find the limit of greatest integer function
f(x) =$\lim_{x->4^{-}}(5[\![x ]\!] -7)$ as x approaches to 4 from left side.
Solution : Since x approaches 4 from the left it must go first through such values as 3.1,3.2,3.3,3.5, 3.9, 3.9999, and so on. But all these values become 3 on the greatest integer function $[\![x ]\!]$. So instead of 4 we must plug in x = 3.
$\lim_{x->4^{-}}(5[\![x ]\!] -7)$ = 5(3) - 7 = 15- 7 = 8

$\lim_{x->4^{-}}(5[\![x ]\!] -7)$ = 8

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