Suppose that liman=0 Prove that limn→∞(1+anxn)n=1
I was trying Leibniz theorem earlier but it was not working. Was I using the right one?
Answer
Using that (1+tn)n→et=exp(t)
as n→∞ with anx:=t you obtain that limn→∞(1+anxn)n=limn→∞exp(anx)=…
Due to the continuity of the exponential function f(t)=et you can interchange the lim operator with the exp operator to conclude that …=exp(limn→∞anx)=exp(0)=e0=1
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