Dying video cards can display a wide range of symptoms. I don't feel sufficiently versed to list them all, but here are some:
CMOS beep code at startup
No display, dark display, washed-out display, flickering display, lines in display, etc
System unable to use higher resolutions/32bit color
Extreme lag or crashes while using video-card intensive applications (like games)
Of course, all these symptoms can also result from other things, like a failing monitor, software issues, memory problems, etc. Perhaps it would be best if you described the problem, and we could go from there?