"Training to failure" can mean several different things.
Technical Failure: This level of failure occurs when you are no longer able to complete more repetitions of the exercise with proper form due to fatigue. You may be able to continue the exercise, but further repetitions will be with compromised form. This failure is usually momentary - you may be able to execute more repetitions with proper form after a short rest.
Muscular Failure: This level of failure occurs when you are unable to continue the exercise even with compromised form. You experience momentary muscular failure, and even with aid or compromised form you cannot continue the set.
Two other terms sometimes appear for failure training.
Positive Failure: You can no longer execute a concentric lift with proper form - for example, you are unable to raise your chin over the bar in a pullup, or unable to lift a barbell from the floor in a deadlift. If you continue to lift past this point - for example, someone helps you raise the weight - you may experience negative failure.
Negative Failure: You can no longer longer execute a controlled negative of the exercise. For example, you may be unable to stop the descent of the barbell in a bench press, so it simply falls to your chest.
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