Let's look at the ultimate extreme version. Assume she's woken up once (or arbitrarily many non-zero times) for tails, and not at all for heads. Now the fact that she's been woken up implies tails with certainty. So if the answer remains 1/2 in the extreme versions, then there must be a discontinuous jump, rather than convergence, when the ratio of the number of awakenings for heads vs. tails tends towards zero.