I am a concerned about this because I am continuing on with a late evening connecting flight in LIM. It's a United flight to IAH, not even a partner with LAN
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Did you buy two separate tickets (how are you getting to LIM, on UA)? Or did your tickets issued by a single entity (such as UA, or through a travel site like Expedia). If the latter, than you are not entirely on your own. If you miss connect, the issuer of the single ticket is obligated to get you to your destination. Not so if different tkts on different carriers (non-alliance).