This is a discussion on $10 NLHE 6-max: AQs vs MP Limp within the online poker forums, in the Cash Game Hand Analysis section; Hold'em No Limit - $0.05/$0.10 - 6 players
UTG: $13.59 (136 bb)
MP: $8.22 (82 bb)
CO: $23.13 (231 bb)
BU: $10.50 (105 bb)
SB: $12.00 (120 bb)
BB (Hero): $10.89
If your goal is to punish the limper in preflop, on the flop you are representing a wider range with your check. And as for your villain, do you really think he wouldn't bet if he flopped a set? The maximum amount combinations of set combos that V can have on the flop is 6 (2 combos for each set) and the total of cards now is 47, which makes a probability of 12.76% that V has obtained a set on the flop.
But actually from a villain's perspective, the chance of landing a set preflop is 6/52 or 11.53%. This indicates that most likely a standard V when he hits a set on the flop, he will try to get value from that point on.
Unless you have observed that this villain is very passive and does not usually bet his bingo hands well (when he should bet), we would not have to include these combinations very often in his range on that board.
On the turn it looks like you are trying to recover value from some hands that you didn't bet on the flop. The question is: What hands can we really bet? We shouldn't having bet hands like KK; QQ; TT OTF? Therefore our range is perceived more as made up of semi bluffs. Maybe gutshots with blockers of heart and of course Axs we have too.
And regarding the calling range V, there could be flush draw combos, in this case it is good not to block the ace of hearts. Now with greater security we can say that if V slowly played in preflop with a strong combination like yours (AQ or AK), he could possibly behave more aggressive here (mainly if his hand is also a flush draw).
Therefore we are winning at hands that have implied river probability and are only beaten by A4hh, as any other 2 pair off combo would seek protection here.
The river is neutral because it really is an absolute brick. However, if we really believe that this villain played passive preflop, now is the right time where he can bluff with a hand like AKo, as draws have failed for us.
Therefore, it seems more appropriate to try a smaller bet, which acts as an action blocker. And fold a raise.
But if your suspicion is that this villain has a wider range, I like your size and would even call to try and catch a bluff.