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Given a function f(x) = lim n->infintiy [x]^n / (x^n +1) , where [x] is a greatest integer function.
What is the limit value of lim x->1+ f(x) ?
Is the limit found above the same with lim n->infinity (lim x->1+ [x]^n / (x^n +1) ) ?
I am rather confused with the above two cases. I don't know how to think of it. I have such kind of thought: if n tends to infinity first, even x becomes very close to 1, the value is still at infinity. If x tends to 1 faster, even n tends to infinity, the value is still 1.
What's wrong with my concepts?
What is the limit value of lim x->1+ f(x) ?
Is the limit found above the same with lim n->infinity (lim x->1+ [x]^n / (x^n +1) ) ?
I am rather confused with the above two cases. I don't know how to think of it. I have such kind of thought: if n tends to infinity first, even x becomes very close to 1, the value is still at infinity. If x tends to 1 faster, even n tends to infinity, the value is still 1.
What's wrong with my concepts?