The Limit As A (Partial) Function

In math, we often say the limit of a function. This is not precise. A better phase is “the limit of a function at a point” since limits are always defined with respect to a function and a point.

Say we have a function \(f\) and we want to find the limit at the point \(p\). We can write \(lim_{x \to p} f(x)\) as \(l(f,p)\) to emphasize that the limit is a binary function, with the following type signature (in Haskell):

lim :: (a -> b) -> a -> b

Except that it’s not quite a function. The limit does not always exist at a point. So perhaps a better type signature is

lim :: (a -> b) -> a -> Maybe b

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