Limiting x->0: Analyzing f(x) = 1/|[[x]]|

  • Thread starter madah12
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In summary, the limit as x approaches 0 of the function f(x) = 1/|[[x]]|, where [[x]] is the greatest integer function, has a left-hand limit of 1 and a right-hand limit of 0. This is because for x values close to 0, the greatest integer function evaluates to 0, making the absolute value equal to 0 and the function equal to 1.
  • #1
madah12
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Homework Statement


Limit x->0 f(x)
where f(x) = 1/|[[x]]|
where [[x]] is the largest integer function

Homework Equations





The Attempt at a Solution


I am having problem redefining the absolute value here
|[[x]]| = -[[x]] when [[x]] <0
so that is when x<0?
the right hand limit is supposed to be infinity
but I am confused about the left one is it 1?
or is it negative infinity or infinity?
 
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  • #2
You are overthinking this. To take the "limit as x goes to 0", look at points close to x= 0. What is [[0.00001]]? What is [[-0.000001]]? What are the absolute values of those?

Do "limit from the right" (x> 0) and "limit from the left" (x< 0) separately.
 
  • #3
it isn't the absolute value function its the greatest integer function
http://www.icoachmath.com/SiteMap/GreatestIntegerFunction.html
but my problem is that since .9999999 continuously is 1 should I consider the greatest integer of -.00000000000...01 as 0 or -1?
 
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  • #4
You said it was "the absolute value of the greatest integer function".

And the greatest integer less than or equal to 0.0000000...01 (where the ... must mean some specific number of missing 0s, not an infinite string) is 0, of course. Since it is positive why would you even consider -1? That has nothing to do with "0.9999999..." being equal to 1.
 
  • #5
the limit from the right is 0 and the left its 1 right?
 
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Related to Limiting x->0: Analyzing f(x) = 1/|[[x]]|

What is the limit of f(x) as x approaches 0?

The limit of f(x) as x approaches 0 is undefined. This is because the function has a vertical asymptote at x = 0, so the value of the function approaches infinity as x gets closer to 0 from both sides.

What is the behavior of f(x) as x approaches 0 from the positive and negative sides?

As x approaches 0 from the positive side (x > 0), the value of f(x) approaches positive infinity. As x approaches 0 from the negative side (x < 0), the value of f(x) approaches negative infinity.

Can the limit of f(x) be determined algebraically?

No, the limit of f(x) as x approaches 0 cannot be determined algebraically. This is because the function has a vertical asymptote at x = 0, so the limit is undefined.

How does the graph of f(x) look like?

The graph of f(x) has a vertical asymptote at x = 0 and approaches positive and negative infinity as x gets closer to 0 from both sides. It has a "V" shape, with the point (0,1) at the vertex of the "V".

What is the significance of the absolute value in the function f(x) = 1/|x|?

The absolute value in the function f(x) = 1/|x| ensures that the function is always positive, regardless of the value of x. It also creates a vertical asymptote at x = 0, resulting in the unique behavior of the function at this point.

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