What is the difference between an asymptote and a hole?

1 Answer
George C.
Nov 7, 2015

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.

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