What is the end behavior of the greatest integer function?

Answer 1

The end behavior is #y=x#. The left end goes to #-oo# and the right end goes to #oo#.

As #x# gets large positively or negatively, the decimal part becomes very small. If you extend the below graph, you should see that it looks like #y=x#. Another name for the greatest integer function is the floor function.

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Answer 2

Xavier Alonso

The greatest integer function, denoted by ⌊x⌋ or sometimes as "floor" function, has end behavior such that as x approaches positive or negative infinity, the function approaches the nearest integer value without exceeding it for positive infinity, and the nearest integer value without being greater than it for negative infinity. In other words:

As x approaches positive infinity, ⌊x⌋ approaches the largest integer less than or equal to x.
As x approaches negative infinity, ⌊x⌋ approaches the largest integer greater than or equal to x.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.