Use hints to make statement obvious
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.