How to find the equation of vertical and horizontal asymptotes?

Please find some specific examples of functions for which you want to find vert. or horiz. asy. and their equations.  This is a broad topic.

Very generally, vert. asy. connect only to rational functions; if the function becomes undef. at any particular x-value, that x-value, written as x = c, is the equation of one vertical asy.

Very generally, horiz. asy. pertain to the behavior of functions as x grows increasingly large (and so are often associated with rational functions).  To find them, we take limits of the functions, letting x grow large hypothetically, and see what happens to the function.  Very often you end up with the equation of a horiz. line, your horiz. asy., which the graph usually (but not always) does not cross.