{"id":505,"date":"2026-09-14T17:54:01","date_gmt":"2026-09-14T16:54:01","guid":{"rendered":"https:\/\/blog.samuelgill.net\/?p=505"},"modified":"2026-09-14T17:54:01","modified_gmt":"2026-09-14T16:54:01","slug":"taking-limits-to-the-limit","status":"publish","type":"post","link":"https:\/\/blog.samuelgill.net\/index.php\/2026\/09\/14\/taking-limits-to-the-limit\/","title":{"rendered":"Taking Limits to the Limit"},"content":{"rendered":"\n

There are many structures in mathematics that turn up in different forms in a variety of contexts. The concepts of Duality or Quotients can be found throughout the mathematical landscape, but there is one which is fundamental to many areas. From Category Theory to Partial Differential Equations, limits are defined everywhere, but often with different definitions, so what is a limit? I’d recommend you read the start of my Baking Topological Doughnut<\/a> post for a refresher on some basic topology, as we’ll be using open neighbourhoods here.<\/p>\n\n\n\n

So, what actually is a limit?<\/strong><\/p>\n\n\n\n

A limit, in the most general sense, can be though of as the object which fills in some gap, so you might classically think of a sequence limit as filling in the gap at the end of the infinite sequence<\/p>\n\n\n\n

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But this can be too restrictive, we can view a limit over the reals<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\lim_{x \to a} f(x)\]\"<\/p><\/p>\n\n\n\n

as the gap at the end of a function, as below.<\/p>\n\n\n\n

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We can always view the first case as the second, by using step functions, but this is too forced. We would like to find some way to extend the second case to the first. <\/p>\n\n\n\n
Let’s look at our second example above, for the sake of our notation, we want to place \"a\" at infinity. How can we do this? We just change variables, we can set <\/p>\n\n\n\n
   <\/span>   <\/span>\"\[x = a(1-e^{-y}) .\]\"<\/p><\/p>\n\n\n\n
Then as \"y \to \infty\", \"x \to a\". Then we view \"f\" as a function of \"y\" instead of \"x\".<\/p>\n\n\n\n
What’s ne(x)t?<\/strong><\/p>\n\n\n\n
In this view, we can find the similarity between our two limits. They both feature two sets, \"\mathbb{N}\" and \"\mathbb{R}\", and some way to tell when a value is later in the set than another, an order \"\le\". An order here is a binary relation (Think \"=, \cap, \text{OR}\") which satisfies the following properties. <\/p>\n\n\n\n