Real limits and the binomial test
Suppose you want to test that your coin is fair using the normal approximation to the binomial test. You flip the coin several times and observe the number of heads. To do the binomial test, first calculate the z-score of the number of heads you observed.
If this z-score is not in the critical region, you should:

Check the limits and see if either is in the critical region. If either one is not in the critical region, fail to reject the null hypothesis.
Fail to reject the null hypothesis.
Reject the null hypothesis.
Check the limits and see if either is in the critical region. If either one is in the critical region, reject the null hypothesis.