Another word for "edge" is "verge." When you verge on something, or when you're on the verge of something, you come close to it, almost about to do it, as if you're right there by the edge of it.
If you're on the verge of laughter, you're not laughing yet, but you're really close to laughing! You might laugh out loud any second now.
If you're on the verge of crying, you're not crying, but you're almost crying. You might go ahead and cry, or you might recover yourself and stop the tears from flowing, like this girl does. To be on the verge of tears is to be really close to tears.
If you're on the verge of shouting or yelling, you're not doing it yet, but you're about to. If you're on the verge of understanding something, you don't quite understand it yet, but you're getting really close! And if you're on the verge of winning, you haven't won yet, but you're about to.
We often talk about people being on the verge of a meltdown, or on the verge of a nervous breakdown. In fact, if you just say you're on the verge, it means you're on the verge of melting down, freaking out, or otherwise panicking.
You can also say that one thing is verging on some quality, meaning it almost has that quality, but not quite. For example, turquoise is a shade of blue verging on green: it's getting pretty close to green.
Remember: to verge on something is to get really close to doing it or being it. And to be on the verge of something is to be almost doing it, but not yet.
Idea 1: "I'm verging on (tears, panic, or frustration) as (something really sad, scary, or difficult happens)."
Idea 2: "It was so difficult for me to learn to (do something); in fact, I was on the verge of giving up, when suddenly I (made progress or had success somehow)."
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