Limits of Function f(x) with Greatest Integer [x]: Problem Analysis

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In summary, the limit value of lim x->1+ f(x) is 1. This is the same as the limit found above, lim n->infinity (lim x->1+ [x]^n / (x^n +1)). However, if n tends to infinity first, even as x becomes very close to 1, the value is still at infinity. If x tends to 1 faster, even as n tends to infinity, the value is still 1. The concept of [x] being the greatest integer less than or equal to x is correct. The limit x->1+ of [x]/(1+x^n) is also 1, as both the numerator and denominator exist and are nonzero for
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abcd8989
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Given a function f(x) = lim n->infintiy [x]^n / (x^n +1) , where [x] is a greatest integer function.
What is the limit value of lim x->1+ f(x) ?
Is the limit found above the same with lim n->infinity (lim x->1+ [x]^n / (x^n +1) ) ?
I am rather confused with the above two cases. I don't know how to think of it. I have such kind of thought: if n tends to infinity first, even x becomes very close to 1, the value is still at infinity. If x tends to 1 faster, even n tends to infinity, the value is still 1.
What's wrong with my concepts?
 
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  • #2
So [x] means the greatest integer less than or equal to x? If so, what is the limit x->1+ of [x]/(1+x^n)?
 
  • #3
Yes, but I m not sure about the limit..I think it should be 1..
 
  • #4
Note that both numerator and denominator exist and are nonzero for all x>1, and thus the limit is rather easy to evaluate.
 
  • #5
as x increases without bound, the greatest integer returns decimal values of n.something back to n, as the denominator keeps growing; approaching infinity.
 
  • #6
In fact, since you want to take the limit as x goes to 1 from above, you only need to look at values of x between 1 and 2.

[itex]1^n[/itex] is pretty easy, isn't it?
 
  • #7
For that matter, try just plugging 1 into the equation.
 

What is the definition of a greatest integer function?

A greatest integer function, denoted as f(x) = [x], is a piecewise function that rounds down any real number x to the nearest integer less than or equal to x. This means that if x is a whole number, the function will return the same value, but if x is a decimal or fraction, the function will round down to the nearest whole number.

What is the limit of a greatest integer function?

The limit of a greatest integer function depends on the value of x. If x approaches a whole number from the left, the limit will be the integer value of x. If x approaches a whole number from the right, the limit will be one less than the integer value of x. This is because the function rounds down, so as x gets closer to a whole number, the function will return the previous integer value.

How do you graph a function with a greatest integer?

To graph a function with a greatest integer, you should first plot the points for the function without the greatest integer. Then, for any x-value that is not a whole number, you should draw a vertical line at that x-value and mark the point where the line intersects the graph. This will create a "staircase" type of graph, with each step representing a whole number value.

What are some common uses of greatest integer functions?

Greatest integer functions can be used to model real-world situations, such as tracking the number of items sold or the number of people in a room. They can also be used in computer programming to round down numbers to the nearest integer.

How can greatest integer functions be used in calculus?

In calculus, greatest integer functions can be used to help determine the continuity and differentiability of a function. They can also be used to solve problems involving limits and derivatives. Additionally, greatest integer functions can be used to construct piecewise functions, which are often used in calculus to model real-world situations.

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