Hi Frank,
because the expression is not defined at x=0. You cannot substitute x->0 because you would divide by zero. The limit that you calculate does exist. If you run:
FunctionDomain[(x + 1/x), x]
You obtain:
x < 0 || x > 0
The function is not defined for x=0. Hence, x=0 cannot be a solution of the inequality.
Cheers,
Marco