couple of ways i know of without removing the shock/strut:
1. drive on the freeway/highway and have someone in another car look at the wheels. if the wheel is bouncing up and down like a dribbling basketball, the shock/strut is blown.
2. push down on any corner of your car to see how easy it goes down and come back up. blown shock/strut means you can push down relatively easy and bounces up rather quickly. this method is harder to tell since springs rates will vary the amount of bounciness.