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Based on villains and situations, I think you need to decide your goal with the hand. Usually with a small pocket pair, it's either to snap off a set and take a large chunk of money from a bigger pair like AA or AK when an ace spikes. On the other hand, in this scenario with just a limper and blinds, you could have played it bigger to chase out the dead money and go heads up with position representing more strength than you actually had. By limping, you're basically trying to go the first route, but no one has shown strength, so it could be more difficult to extract money if you do hit.
I am currently playing at these stakes and there are MANY villains who typically play tight/passive/aggressive who LOVE to shove all-in when they have a big hand because someone will pay them off. If I had a read on that villain as such a player, I could see myself laying it down with just the two of us in the hand, although with an Ace on board it's a little tougher as you wonder if they'll do the same thing with AK. On the other hand, if it's a looser, weak player, it's pretty difficult to lay that down. I actually think the way this hand played out, it was theoretically easier for you to get out because the villain pushed all-in after a bet and not one, but TWO re-raises. Unless he is really loose and clueless, what hand is he going to push in with that you can beat?
But, to get back to the point on chasing hands out or not, it goes back to how you want to play it. With pocket 3's on the button, you have the flexibility to go either route and then you have to play well post flop to take advantage. So, I don't think it was necessarily a mistake to limp, but I think you might think it was a mistake in hindsight since you know the result.
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