Unfortunately it's hard to say if a projector lamp has failed, unless of course, it has exploded. But, there are a few things that you can check yourself. When the projected image has become less and less it's a sign the projector lamp has reached the end of its life. When the light output is very low or when there is no image projected at all, you can take out the projector lamp and examine it while looking at it from the front. In the center of the lamp you will see a glass part, the strike stem or burner that contains two electrodes. When the burner is not aligned in the center of the lamp anymore it has failed. When you see white or black spots on the burner it has most likely also failed. If you are unsure you are always welcome to take some pictures of the lamp (and the burner especially) so we can take a look for you.
The strike stem is misaligned and has white spots. If the stem is not clear it's a sign of lamp failure.