Use hints to make statement obvious
I don't like the hint, and don't like the solution. The statement is a little tricky and it may not be clear to everyone what it asks for. One of the hints should be to negate the statement. And the hints or solution should talk about the cases when the limit is 0 and +/- infinity. Once this is all laid out properly the question will be trivial. Why not choose f=g=1 to drive to point home? And only then give a more sophisticated example? I also don't think using constants a,b,c,d is doing anything for us, plug in numbers and everyone will be able to generalize. Also, in the solution I see p/q, then f/g and then f=f/g, please, let's make this cleaner.
I have to be honest, I like the hint. I think the hint makes the question seem more real to students. Can a function has a horizontal asymptote not at 0? Well of course it can! But then wait can it have a horizontal asymptote not at 0 if we divide by two polynomials and not take just any function?
But I do agree the example is severe overkill - I don't think the Mobius transformation is a good idea for a solution to problem one on an exam...
I will fix the solution - but I don't think I can fix the hint partially because I agree with it and partially because I wouldn't be sure how else to change it.
The new first hint is even better than I was asking for, good job. Also the solution is much improved. I added an example of a non-constant polynomial, just to point out that the easy way out is by far not the only type of counter example.