I understand how to solve the problem but how is the answer even possible?

I too didn't expect the answer to be 1/2 but I can't think of a reason why the OP thinks that the answer should have been infinity. Just visualizing or drawing the graphs you see that the limit will not be infinity--in fact it will not be a 'large' value.
That isn't entirely unreasonable; the perpendicular bisector is approaching vertical, and a vertical line would intersect another vertical line "at infinity", as one might imagine it. We who have more experience visualizing loci can avoid that expectation, but even we can see things incorrectly in unfamiliar circumstances. And keep in mind the curse of expertise.
 
And since beating on a dead horse is my second name, here is an illustration of another similar limit, this time from intersections of the normals with the Y axis:

View attachment 32778
Still beating :) : this actually shows that the curvature radius is 1/2, which means that the curvature is 2 (=d2dx2x2)\left(= \frac{d^2}{dx^2} x^2 \right)
 
Two parallel lines never meet, but these lines aren't quite parallel and as they tend towards being parallel they also become very close to each other (in the limit as P approaches the origin). Of course when P is AT the origin there's no bisector.
 
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