Assume observations are from a subclass of a one parameter exponential family whose dominating measure is nonatomic. Consider a one-sided sequential testing problem where null and alternative parameter sets have one common boundary point. Let the risk function be a linear combination of probability of error and expected sample size. Our main result is that a sequential test is inadmissible if its continuation region has unbounded width in terms of the natural sufficient statistic. We apply this result to prove that weight function tests, with weight functions that contain the common boundary point in their support, are inadmissible. Furthermore any obstructive test is inadmissible, where obstructive means that the stopping time for the test does not have a finite moment generating function for some parameter point. Specific tests of the above type are cited.