How can I find limn→∞(1+xn)√n?
I know limn→∞(1+xn)n=exp(x) but I don't know how can I put the definition in this particular limit.
I know then, that limn→∞(1+xn)=1, but I don't think this is right to consider.
Answer
limn→∞(1+xn)√n=limn→∞[(1+xn)nx]xn√n
From
limn→∞[(1+xn)nx]=eandlimn→∞xn√n=0, **, we get
limn→∞(1+xn)√n=e0=1
EDIT
I add the note bellow as my calculation was considered insufficiently justified
**and because the terms are positive, and we don't have an indeterminate case 00 or 1∞ or ∞0,
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