Limit of the greatest integer number

Thread starter LikeMath
Start date Jan 27, 2012
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Integer Limit

In summary, the greatest integer function, denoted as ⌊x⌋ or [x], rounds a real number down to the nearest integer. Its limit, also known as the ceiling function, is the largest integer that is less than or equal to the given real number. The limit can be calculated by subtracting the decimal part of the given number. As the input approaches infinity, the greatest integer function has a horizontal asymptote. It is used in various mathematical applications and can also model real-world situations.

Jan 27, 2012

LikeMath

Hi there,

It is clear that

[itex]\lim_{n\to\infty }\frac{1}{n}\left[\frac{n}{3}\right]=\frac{1}{3}[/itex].
But the problem that I could not get a formal proof!

Thank you.

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Jan 27, 2012

micromass

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Try the squeeze theorem.

Jan 27, 2012

LikeMath

Yes, Thank you.

Related to Limit of the greatest integer number

1. What is the greatest integer function?

The greatest integer function, denoted as ⌊x⌋ or [x], is a mathematical function that rounds a real number down to the nearest integer.

2. What is the limit of the greatest integer function?

The limit of the greatest integer function, also known as the ceiling function, is the largest integer that is less than or equal to the given real number. For example, the limit of the greatest integer function of 3.4 is 3.

3. How is the limit of the greatest integer function calculated?

The limit of the greatest integer function can be calculated by taking the given real number and subtracting its decimal part. For example, if the real number is 3.9, the limit would be 3 (3.9 - 0.9 = 3).

4. What is the behavior of the greatest integer function as the input approaches infinity?

The greatest integer function has a horizontal asymptote at infinity, meaning that as the input approaches infinity, the output remains constant at the largest integer value.

5. How is the greatest integer function used in real-world applications?

The greatest integer function is used in many areas of mathematics, including number theory and calculus. It can also be used to model real-world situations, such as calculating the maximum number of items that can fit in a certain space or determining the highest possible score on a test with a set number of questions.