All right, so sorry to hear that some lime juice have splashed into your eyes. As a matter of fact, that stuff is highly acid and might be irritative to your eyes, and make you feel very uncomfortable instantly. Thus, you should try to get plenty of water to flush your eyes, afterwards, you gotta try to get some eye drops to help. Try to get some medical help when necessary and never rub your eyes!!
Take it easy. Since your eyes are very vulnerable and sensitive, so when lime juice enters your eyes, you may feel pain. As long as lime splashes to your eyes, they must water immediately. That's the protection your eyes have for themselves. All you have to do is to wash your eyes as soon as possible to get the lime juice out of your eyes. You will be OK. It's not big deal.
If your eyes have some problems themselves before this, the situation may last or get worse, then you need go to see a doctor.
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