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  "body": "\n## Summary\n\nA filter limit is the limit of a function $f \\colon S \\to X$ along a [[filter|15PF73BKY6]] on $S$, where $X$ is a [[topological space|0CQHFPGJ45]].\nThe function tends to $x$ along the filter when the preimage of every [[neighbourhood|0ZCY0Z80B6]] of $x$ belongs to the filter.\nOne notion covers limits of sequences, limits of functions at a point or at infinity, one-sided limits and sums of families, by choosing the filter.\nIt is the same as the [[convergence|70M1VAPFN0]] of the image filter $f(\\mathcal{F})$ to $x$.\n\n## Explanations\n\n### L1 human\n\nA filter on $S$ describes a way of \"going to the limit\" in $S$, by saying which sets are eventually entered.\nFor sequences, the sets that are eventually entered are the sets that contain all $n$ from some point on.\nA function has limit $x$ along the filter if, for every small region around $x$, the points where the function lands in that region form an eventually entered set.\nWith the filter of tails of $\\N$, this is exactly the usual limit of a sequence.\n",
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