Question

What are the asymptote(s) and hole(s), if any, of `f(x) = tanx`?

Answer 1

`f(x) = tan(x)` is a continuous function on its domain, with vertical asymptotes at `x = pi/2 + npi` for any integer `n`.

Explanation:

`f(x) = tan(x)`

has vertical asymptotes for any `x` of the form `x = pi/2 + npi` where `n` is an integer.

The value of the function is undefined at each of these values of `x`.

Apart from these asymptotes, `tan(x)` is continuous. So formally speaking `tan(x)` is a continuous function with domain:

`RR "\" { x : x = pi/2+npi, n in ZZ }`

graph{tan x [-10, 10, -5, 5]}