What is a vertical asymptote and where do they occur for a cosecant ?

Question

What is a vertical asymptote and where do they occur for a cosecant ?

Answer 1

cosecant = 1/ sine

1/sin(x) becomes infinite when sin(x)=0

sin(x)=0 when x=0 or x=PI

x=0 is the vertical asymptote

x=PI is also a vertical asymptote

Answer 2

A vertical asymptote is a vertical, or up and down limit which a function approaches, but never actually reaches, because it is not allowed to assume that value.  Usually this is due to prohibition against dividing by 0.

The cosecant function is the reciprocal of the sine function.  So, whenever the sine function approaches 0, the cosecant function approaches the vertical limit of (1/ sin Θ) = (1/0) = ∞.  The sine function equals 0 at Θ = kπ, where k is any positive or negative integer.  These are also the vertical asymptotes of the cosecant function.

So, for the cosecant function, the vertical asymptotes are at kπ, where k is any positive or negative integer.