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Notes -
I'm going to defer to the mathematicians on what it is "for", but analysis is one of the major fields of modern mathematics - you need limits to do almost anything rigorous with general real numbers. I agree the only thing you explicitly use limits for in high school maths is making calculus look rigorous, but other very obvious applications in "maths for physicists" are summing infinite series and explaining what it means for numerical algorithms to converge. But limits come up everywhere - in frequentist statistics probability is a limit.
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