Show that for any r>0, lnx=O(xr) as x→∞
I know that if xn=O(αn) then there is a constant C and a natural number n0 such that |xn|=C|αn| for all n≥n0. But in this case I do not have sequences, how can I work with these functions? In this case there would be no natural number? Would only the constant be demanded? One would not have to lnx≤x for all x>0 and with this could not solve much of the problem with C=1?
Answer
If you can use derivative-based methods:
limx→∞lnxxr=limx→∞1/xrxr−1=limx→∞1rxr=0
No comments:
Post a Comment