Prove lim as x--> infinity of (lnx)/x=0 by first showing

[msl1333]

Prove that:

lim as x--> infinity of (lnx)/x=0

by first showing that for x>1 lnx= integral from 1 to x of dt/t is less than or equal to the integral from 1 to x of dt/(t^.5)=2*((x^.5)-1).

Hint: Compare the graphs of y=1/(t^.5) and y=1/t for t greater than or equal to 1.

I feel like intuitivley I can understand the reasoning, but I'm not sure how to explain it. Any help would be appreciated!

 

[pka]

\(\displaystyle x > 1 \Rightarrow \,\ln (x) < x\)

\(\displaystyle \ln (x) = 2\ln \left( {\sqrt x } \right) < 2\sqrt x\)

\(\displaystyle \frac{{\ln (x)}}{x} < \frac{{2\sqrt x }}{x} = \frac{2}{{\sqrt x }}\)