What is the limit of the greatest integer function?

The limit of the greatest integer function, often denoted as ⌊x⌋, is a bit tricky because the greatest integer function is not continuous. The greatest integer function, also called the floor function, gives the greatest integer less than or equal to a given real number x.

For example:

⌊3.7⌋ = 3
⌊-2.3⌋ = -3
When it comes to limits, we need to consider the behavior of the function as the input approaches a particular value.

Limit at a point:
If we are trying to find the limit of the greatest integer function as x approaches a specific value, we need to examine the behavior from both sides (left-hand limit and right-hand limit).

Let's say we want to find the limit of ⌊x⌋ as x approaches some value c.

Right-hand limit (x → c⁺): This is the limit of ⌊x⌋ as x approaches c from the right (i.e., x > c but very close to c). Since the greatest integer function steps down to the next integer as x crosses each integer value, the right-hand limit will be the greatest integer less than or equal to c.

Left-hand limit (x → c⁻): This is the limit of ⌊x⌋ as x approaches c from the left (i.e., x < c but very close to c). In this case, the left-hand limit will be the greatest integer less than or equal to c.

Since the greatest integer function has a jump at every integer, the right-hand and left-hand limits at any integer point will differ. Therefore, the limit does not exist at integer values.

Example:
If we want to find the limit of ⌊x⌋ as x approaches 2, we have:

Right-hand limit: lim(x→2⁺) ⌊x⌋ = ⌊2.1⌋ = 2
Left-hand limit: lim(x→2⁻) ⌊x⌋ = ⌊1.9⌋ = 1
Since the right-hand and left-hand limits are not equal, the limit of ⌊x⌋ as x approaches 2 does not exist.

Conclusion:
The greatest integer function has no limit at integer points because of the discontinuous jump at each integer. The limit of ⌊x⌋ exists only at non-integer points, where the function value remains constant as x approaches that point. However, at integer points, the limit does not exist due to the discontinuity.