In what situations would I want to use a polarized vs. a merged range? Are there situations where it is best to have a purely high or low value range?
When to use a low-value range
There is no time when an exclusively low value range is optimal. The purpose of including low value hands in a range is to extract additional value from high value hands, or to play more hands in general against opponents that fold too often. In the first case, you need high value hands in your range to make the strategy work. In the second, there is no reason to limit yourself to low value hands.
When to use a high-value range
Against loose-passive opponents, who will play most hands and take them to the river, you will want to play high-value hands. Since these players can't be bluffed, you need hands that can get to showdown and win. Your "high value" range can expand to include some medium hands depending on how light opponents will call down (if they call down with any pair, hands that make second pair can be played profitably), but it will not include low value hands.
This is the best strategy against a level 0 player.
When to use a polarized range
As your opponents become more able to fold hands in certain situations, it becomes profitable to start adding bluffs to your range. When adding bluffs to a range, it is best to start with hands that have little-to-no showdown value - the 72o-type hands. There are a few spots this can be applied. A polarized range can be used when 3-betting in position pre-flop against a good opponent. A good opponent will not call with many hands out of position - they may have some high value hands in their calling range, but for the most part they will 4-bet or fold. In this situation, you will either want to get your money all-in, or fold. You go all-in with your high value hands, and fold the rest. So if you 3-bet with a hand like AJs, but have to fold to an all-in, that is a waste of a hand that can be profitably played in position. This is why we polarize our range - when we get 4-bet with 72o, and have to fold, we aren't losing value.
A similar place where a polarized range can be useful is a flop check-raising situation. When the board comes out K 8 4 rainbow, and you check-raise, you are representing a very strong range - mainly sets, possibly AK. To balance that range, you can add hands like small pocket pairs or missed suited connectors - these hands have little-to-no value against the range that your opponent will continue with.
This will be the best strategy against a level 1 or level 2 player.
When to use a merged range
A merged range comes into play when your opponent starts adjusting to your polarized range. On the previous flop of K 8 4, an opponent may start adjusting to your polarized raising range by calling down with second pair, or a pocket pair like 99 or 77. Suddenly, we can start raising hands like KJ and QQ for value. A merged range will put opponents to many more difficult decisions, which will likely lead to them making more mistakes.
This strategy will be best employed against higher level thinkers, where there are significant meta-game factors and constant adjustment of ranges as the match proceeds.
One example I can come out is a situation where if you make a raise, you put your opponent to a decision that they will be comitted whatever they do, so either they need to push or fold. Because of this reason, you have the same decision too after they made their (push or fold). So getting in a situation like this with a range not polarized is simply bad, because if you are comitted, you have to call the push if you have the right odds, and put more money in bad... but if you fold a mid-strong hand, you wasted your potential to win the hand by calling. Example: you raise the turn with a middle pair or strong draw such amount that your opponent can't just call. In this case you lose nothing with zero equity hand when your range is polarized. You would lost with that hand anyway, and you can make a -EV situation profitable if they fold enough, but if they push, you don't have to put more money in bad. With a middle strong holding it could be a terrible idea, because you lose the possibility to win at showdown let's say with mid pair, you waste your chance to win if you do this.
You can balance your polarized range perfectly anyway by the correct ratio of good/bad hands in that situation.
Another example came to my mind, when your opponent has polarized range on the river!! There is no point to bet your mid strong hands, because either they will fold their nothing or bluff your not-good-enough-for-calling hand out of the pot with a raise and you could win with a check anyway. And if you bet your hand and still call, it's not sure if the pure bluff enough to make the call profitable, so you are screwed. :D A typical and very common situation when you think your opponent had a draw all the way, and the river card completed no draw at all. Betting with one-pair in this situation is terrible, even some strong but tricky hand too, especially against opponents who can balance their ranges well, so they can check-raise bluff you out of the pot optimal times...
I remember a hand (actually two :D) played by Daniel Negreanu in a high stakes heads up match, where he check-called three times with top pair and good kicker OOP, and then, a similar situation came, and the guy checked behind top pair with good kicker, because he was afraid that Negreanu can call him with a better hand ! He would clearly bet that hand for value without the history (the previous hand) as Negreanu pointed out in his analysis. So, from the opponent perspective, there were no value in merging his value range against Negreanu in that situation.