See the graph I've attached.
At x = 1, you can see that the graph isn't defined. As you approach x = 1 from the right, the graph tends towards positive infinity, and as you approach x = 1 from the left, the graph tends towards negative infinity.
The graph does not tend towards a specific value as x tends towards 1. The value it tends towards is undefined, and hence the limit does not exist.
* right hand side, ln x --> 0 from above (and is therefore small but positive) and hence 1/ln x --> +oo.
* left hand side, ln x --> 0 from below (and is therefore small but negative) and hence 1/ln x --> -oo.
Hence the right hand limit is different to the left hand limit and so the limit doesn't exist.