A dying video card (laptop or otherwise) can be difficult to diagnose, because the symptoms aren't necessarily specific to video cards. They can often be indicative of other problems, so before writing off your video card as dead or dying, make sure you eliminate other factors that could be affecting your video performance. Most notably are out-of-date drivers, programs running in the background, and dust buildup.
One sign of a dying video card is overheating; over time, hardware stops running as efficiently as it once did. Ensure proper airflow by cleaning dust and other buildup from the laptop's interior. If the card is still running very hot (and you're not overclocking), it may be reaching the end of its life.
If your system randomly freezes or crashes during operation, (especially when performing video-intensive tasks) it may be the fault of the video card. If you get a blue screen of death (BSoD), the driver at fault is commonly listed; if it's the video driver, it may indicate a dying card (or a bad driver -- make sure you update).
If games or movies that performed fine in the past now run sluggishly or not at all, your video card may be dying. Your drivers could also be out of date, so be sure to update them. You need to eliminate any other potential factors, such as hardware changes or programs running in the background.
The most obvious sign of a video card being dead or dying is the video not working at all. When you boot your computer and see nothing on the screen, or if your operating system can no longer detect your specific video card (defaulting to a generic, nondescript driver), then your video card may indeed be dying.
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