What is the difference between an asymptote and a hole?

Wendy Larsen

Wendy Larsen

Answered question

2023-01-03

What is the difference between an asymptote and a hole?

Answer & Explanation

hachatpg9

hachatpg9

Beginner2023-01-04Added 14 answers

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)={0ifx=(2k+1)π2forsomekZtan(x)otherwise
Then t(x) has vertical asymptotes at (2k+1)π2 for all kZ, but has no 'holes'.
The function f(x)=x21x1 has no asymptotes, (unless you count y=x+1), but it has a 'hole' at x=1, where f(x) is not defined.

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