It's actually worse than that. Even knowing the very first term of the continued fraction result would already tell you that the number is at least 2. So the computation will hang without ever producing even a single term.

This is a fundamental problem shared by any form of exact real arithmetic with unique representations; you can always then be forced to make a choice, before emitting any information, that depends on infinite amounts of information about the operands. To solve this problem, you would need to choose a representation with redundancy. (With redundancy, equality comparison becomes harder, and it can still diverge, but you can still get approximations of the answer.)