Limit Question: Calculator Discrepancy

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SUMMARY

The limit of the function t(x) = (1 + [1/x])^x as x approaches 0 from the right (x → 0+) is definitively 1, despite the function exhibiting complex values and undefined behavior for x approaching 0 from the left. The graph indicates a gap between -1 and 0, where real values approach 1, but the limit does not exist overall due to the discontinuity. The one-sided limit confirms that while the function approaches 1 from the positive side, it fails to converge from the negative side.

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Question: Why does my calculator state the limit of the following function is 1, but my calculations state it does not exist?

There's a gap in the graph between -1 and 0. I noticed that there are a few points in that section that do generate real values which appear to move towards 1, but is this enough information to presume the limit is 1 at 0?


t(x) = (1 + [1/x])^x

lim t(x) = ??
x->0


Chart:
x-value____t(x)-value
1_________2.000000
.5________1.732051
.1________1.270982
.01_______1.047233
.001______1.006933
.0001_____1.000921
.000001___1.000014
-.000001___0.999986 - 0.000003i
-.0001_____0.999079 - 0.000314i
-.001______0.993112 - 0.003120i
-.01_______0.954617 - 0.030000i
-.1________0.763453 - 0.248061i
-.5________-i
-1_________undefined
 
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What is true is that the one sided limit as x → 0+ is 1.
 

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