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lim (x+1)^x/x^x= lim [(x+1)/x]^x= lim [x/x + 1/x]^x= lim (1 + 1/x)^x
as x-> infinity, 1/x= 0, therefore lim (1 +1/x)^x = (1 + 0) ^infinity= 1^infinity = 1
since 1 power to anything always equals 1
i think you missed something but according to my calculations the answer is e
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you actually cannot eliminate 1/x as it is not seperate
the answer to 1.0000001^1000000000000 equals 2.792 , try it
TI-89 also shown e, but at least you should show him how you solved this
as lim (x+1)^x / (x)^x as x->+infinity becomes inf/inf form as x goes to inf so we use L hopitles rule but the derivative still does not give us th required results as it too remains inf / inf so we use series expansion which is
e+ e/2x +11e/24x^2.....
as the terms with x in denominator get 0 as x->inf so we are left with e