Finding limit- greatest integer function

In summary, the question is whether the limit of x times the floor of 1/x as x approaches 0 is equal to 1. Through formal inequalities, it is shown that the limit is indeed equal to 1. The person asking confirms that this solution is correct and thanks Bhaskar for the help in presenting it formally.
  • #1
life is maths
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Hi, I was studying for my upcoming calculus exam and couldn't be sure if I could solve this question:

[itex]\stackrel{lim}{x\rightarrow0}[/itex] x [[itex]\frac{1}{x}[/itex] ]

If x approaches 0 from left, then 0< x [[itex]\frac{1}{x}[/itex] ]<1

If x approaches 0 from right, then x [[itex]\frac{1}{x}[/itex] ]>1 since x [[itex]\frac{1}{x}[/itex] ]=[itex]\frac{x}{x-1}[/itex]

Hence, the limit is equal to 1.

Is this solution true? Thank you for any help :)
 
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  • #2
we know that t-1< [t] <= t.
Replace here t by 1/x.
I think your answer is correct.This form of inequality is more formal method.
 
  • #3
Thanks, Bhaskar. You're right, I should be able to write it formally.
 

What is a greatest integer function?

A greatest integer function, also known as floor function, is a mathematical function that rounds down any real number to the nearest integer. For example, the greatest integer function of 2.8 is 2, and the greatest integer function of -3.2 is -4.

How do you find the limit of a greatest integer function?

To find the limit of a greatest integer function, we need to approach the given point from both the left and right sides. If the values from both sides approach the same integer, then that integer is the limit of the function. Otherwise, the limit does not exist.

Can the limit of a greatest integer function be a non-integer?

No, the limit of a greatest integer function can only be an integer because the function itself rounds down to the nearest integer. Therefore, the limit will always be the same integer as the given point.

What is the difference between a greatest integer function and a ceiling function?

A greatest integer function always rounds down to the nearest integer, while a ceiling function always rounds up to the nearest integer. For example, the ceiling function of 2.3 is 3, while the greatest integer function of 2.3 is 2.

Can the limit of a greatest integer function be undefined?

Yes, the limit of a greatest integer function can be undefined if the values from both sides approach different integers. This means that the function does not have a single limit at that point.

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