Question

What is the difference between an asymptote and a hole?

Answer 1

The two concepts are quite different and only sometimes coincide.

See explanation...

Explanation:

A vertical asymptote usually corresponds to a 'hole' in the domain, and a horizontal asymptote often corresponds to a 'hole' in the range, but those are the only correspondences I can think of.

For example, we can define the function `t` as follows:

`t(x) = { (0, "if " x = ((2k+1)pi)/2 " for some " k in ZZ), (tan(x), "otherwise") :}`

Then `t(x)` has vertical asymptotes at `((2k+1)pi)/2` for all `k in ZZ`, but has no 'holes'.

The function `f(x) = (x^2-1)/(x-1)` has no asymptotes, (unless you count `y = x + 1`), but it has a 'hole' at `x=1`, where `f(x)` is not defined.