Guild Wars 2

I am level 40 and just ran my first dungeon, AC. I was told afterwards by my guild that the dungeon can only be run once a day. Is that true? Or can you only run one specific path of a dungeon per day?

If there are limits to how many times you can run a dungeon, is there somewhere on my hero screen that I can see what dungeons I’m ineligible for?

There’s no limit to how many times you can run a dungeon, only the limit of your tolerance of the rewards.

Rewards follow these rules:
-On your initial clear of a specific path of a day (see below), you will get full rewards.
-Subsequent clears of that same path on the same day will give you reduced rewards.
-Clearing several dungeon paths in under an hour will result in “diminished rewards” on your rewards. You can remedy this by taking a break. (My group alternates to fractal runs to pass time while we wait for diminishing returns to wear off)
-Full dungeon rewards reset on a new day (4PM PST)

Here’s an example using AC Tokens:
-You clear AC path 1 for the first time of the day (60 tokens)
-You do path 1 again on the same day (20 tokens)
-You do path 2 on the same day (60 tokens)
-You do path 1 again on the same day (20 tokens)

Basically, your rewards are tied to your path progress of the current game day. It’s basically full initial rewards or reduced repeat rewards. Diminished rewards only shows up if you complete the runs very quickly in under an hour.

(If you’re curious, the diminished rewards are a safety net that attempts to discourage exploiters until a better solution is found.)

There is no limit as to how many times you can run a dungeon, the reward does get less the more you run it you will get less tokens then.

every time you become eligible for a new dungeon you will get a mail from your herald to tell you about it and mark it on your map

http://wiki.guildwars2.com/wiki/Diminishing_returns
no limit but….
" DR will trigger if the dungeon is completed consecutively and/or in under 30 minutes. This obviously affects teams that can clear a certain dungeon mode rapidly. DR will trigger if players cycle between two dungeon modes, even if the dungeons are different. A simple method to avoid this is to do all three explorable paths in order. DR affects all rewards for completing the dungeon and any chests that spawn in the dungeon. For every point of DR, dungeon rewards will be cut in half until reaching a minimum value."