#tan(x)# is undefined at certain points:
#pi/2,(3pi)/2,(5pi)/2#, and so on.
Graphing #tan(x),# we find that, where #h# is a point where #tan(x)# is undefined:
#lim_(xrarrh^+)tan(x)=oo#, and #lim_(xrarrh^-)tan(x)=-oo#
We'll find the first three vertical asymptotes.
So here, #f(x) " DNE"# when #((5pi)/4-x)rarrh#.
Taking #h=pi/2#, we have:
#((5pi)/4-x)rarrpi/2#, which we can write as:
#(5pi)/4-x=pi/2#
#-x=pi/2-(5pi)/4#
#-x=-(3pi)/4#
#x=(3pi)/4#
Taking #h=(3pi)/2#, we have:
#(5pi)/4-x=(3pi)/2#
#-x=(3pi)/2-(5pi)/4#
#-x=pi/4#
#x=-pi/4#
For #h=(5pi)/2#, we have:
#(5pi)/4-x=(5pi)/2#
#-x=(5pi)/2-(5pi)/4#
#-x=(5pi)/4#
#x=-(5pi)/4#
This series continues to the left and right.