Limits of trig functions as x approaches infinity?

yes, you are right
because of oscillation, usually limits at infinity don't exist
But I think anything is possible, depending how you define the rational function

In the X direction yes. sin and cos oscillates on the x direction for an infinite amount of times. On the Y it is impossible, since it must oscillate at a certain number Y and drop down or up. Thus no longer infinite. However, it could snag and stop, which could create a straight line upwards infinitely.

Since you're asking about the x direction it will almost always be going infinitly + and -.

This is considering the limitations of the graph. If the graph spans long enough it will not be infinite and will snag, but it usually continues along a parabollic path infinitly until another snag.