What is the limit of ln x / x

Since you are approaching it from the positive side you should not use L'Hopitals rule. 
You should instead use very small values of x to see the trend 

if you look at x=.0001, then ln(.0001)/(.0001)= -92103.4 
where as x=1, ln(1)/1=0 
as you can see as x approaches 0, the answer goes to - infiniti