Your thinking is on the right track but there is just one small mistake:

The degrees are in fact equal, because of the square root. In the denominator, as x approaches infinity or negative infinity, the term matters less and less and eventually not at all, as it gets swamped by the increasingly large term, which is square rooted. So essentially, as x approaches positive infinity, the function behaves like . At negative infinity, this is similar, except the numerator will be negative while the denominator will always be positive, so it approaches -1. This accounts for the horizontal asymptotes.because the degree of the denominator is bigger than the numerator

And there are actually vertical asymptotes at , you were correct there.