Monday, 23 June 2014

calculus - Show that for any r>0, lnx=O(xr) as xtoinfty



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 nn0. 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 lnxx 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:
limxlnxxr=limx1/xrxr1=limx1rxr=0


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