Next determine the expected number of turns given such barriers using the same method of interlocking expected values as in the ant and spider problem. The answer turns out to be 2x.
Thus, when x is infinity the expected number of turns must be 2*infinity=infinity.
Related question: What is the expected number of turns it would take a random walker on a number line to revist any point? The answer is here.
Michael Shackleford, ASA