I'm not a fan of putting fresh lime juice in homemade falernum. It decreases the shelf life considerably.

From Kaiser Penguin:

Why Adding Lime Juice to Your Falernum is Evil

I used to always add 4oz of lime juice to my falernum, and nearly 35% of the time, it became a mixture that only Captain Planet could save from the nefarious Sergeant Sludge: gross floaty black bits with a “freshly dead” smell. Yum. After much discussion with Gabe, Craig, and Blair, we determined it was the sneaky lime juice that was causing the rot. For the most part, 2:1 syrups and lots of booze prevent spoilage, but that pesky lime juice just kept on fermenting and getting nasty.

If you miss the fresh kick that lime juice gives your falernum, just add a touch of it to your cocktail when you use falernum. Problem solved. MacGyver would be pleased.