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

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SUMMARY

The limit of (sin[floor(x)])/x as x approaches 0 does not exist. For x approaching 0 from the positive side (lim(x->0+)), the limit evaluates to 0, while for x approaching 0 from the negative side (lim(x->0-)), it approaches -∞. This discrepancy indicates that the two one-sided limits are not equal, confirming the non-existence of the overall limit. The floor function, denoted as [x], plays a crucial role in this analysis.

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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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