Does the Limit of (sin[floor(x)])/x Exist as x Approaches 0?

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around the limit of the function (sin[floor(x)])/x as x approaches 0. Participants explore whether this limit exists and, if so, what its value might be. The conversation includes theoretical reasoning and mathematical analysis.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions whether the limit exists and suggests it might be 0, seeking clarification.
  • Another participant agrees with the idea that the limit could be 0, referencing a method involving delta and epsilon.
  • A different viewpoint indicates that for x < 0, the limit approaches -∞, while for x > 0, it approaches 0, suggesting differing behaviors based on the direction of approach.
  • Another participant argues that the limit does not exist because the limits from the left and right do not match, specifically noting that lim(x->0+) sin(floor(x))/x = 0 and lim(x->0-) sin(floor(x))/x = ∞.

Areas of Agreement / Disagreement

Participants express differing opinions on the existence of the limit, with some suggesting it may be 0 while others assert that it does not exist due to the differing limits from either side of 0.

Contextual Notes

Participants clarify that [x] refers to the greatest integer less than or equal to x, also known as the floor function. The discussion highlights the importance of the behavior of the function as x approaches 0 from both the positive and negative sides.

phymatter
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does lim x->0 (sin[x])/x exist ? if yes then what is it , iguess 0 , but cannot figure out the reason .. pl. help...
note: [x] is greatest integer less than or equal to x
 
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hi phymatter! :smile:
phymatter said:
does lim x->0 (sin[x])/x exist ? if yes then what is it , iguess 0

yes :smile:

(just choose your delta to be 0.9, whatever your epsilon :wink:)
 


If-1 < x < 0, [x] = -1, if 0 < x < 1, [x] = 0. As a result , for x < 0, the limit is -∞ while for x > 0, the limit is 0.
 


phymatter said:
note: [x] is greatest integer less than or equal to x
It's usually called the "floor" function.

I don't think it would exist. lim(x->0+) sin(floor(x))/x = 0 and lim(x->0-) sin(floor(x))/x = ∞. For there to be a limit, lim(x->0+) sin(floor(x))/x and lim(x->0-) sin(floor(x))/x must be equal, they're not.
 

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