Locally geodesic but not quasi geodesic

Question

A path p is called a K-local geodesic if for all x and y in p, $d_{p}(x, y) \leq$ K implies that $d_{p}(x, y) = d(x, y)$.

1. For all K, give an example of an infinite K-local geodesic in Z⊕Z which is not a quasigeodesic.
2. Let X be a geodesic metric space. Assume triangles in X are δ-thin. Show that if an path p in X is an 8δ-local geodesic then it is a (2, 0)-quasigeodesic.
3. Show that if every infinite local geodesic in a metric space X is a quasigeodesic then X is hyperbolic.
   
   I am thinking about all these parts for a while, but haven't been able to figure it out. Any hints or comments are welcome.