"Push yourself beyond your limits" is an often used sound bite on fitness posts. The message should be to push beyond your MENTAL limits in order to nudge yourself to the brink of your current physical capabilities. That's how you progress.
What it should not mean is to participate in exercises way beyond your current tolerance level (e.g. running a 5k just because it's the shortest race when you haven't run a block; Doing 3 sets of 10 just because tradition has deemed that to be where to start when you've been doing 0 sets of 0). Stuff like that leads to prolonged inflammation (soreness) and injury. It's also becomes a mental deterrent to continue on.
It should also not mean doing a few extra cheating reps just to hit a number (e.g. launching/swinging weights around and counting numbers). These mindless reps are valueless and even detrimental (And that's assuming the initial "good reps" had value.
Finally, it definitely does't mean feeling like crap (e.g. nausea, aching joints). The point of the workout is to build off the previous one. One workout does nothing like eating one healthy meal does nothing. It's a commitment, but one that's well worth your time.
Pushing yourself is necessary, but with reason. When you are challenging yourself near the brink of your maximal tolerance it's already incredibly tough. Most get no where near that at it is.
No matter how fast or how hard you try if you can only do 10 pushups, for example, you're not going to will your body to do 20 without diminishing returns. However, you will get there in a safe and reasonable way with time and mental commitment.