For what values of x, if any, does #f(x) = tan((17pi)/12+4x) # have vertical asymptotes?

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Answer 1 · 2016-02-08T16:50:50

#x=pi/48+pi/4k# with #k# an integer.

#tan# has vertical asymptotes when the argument of the function is an odd multiple of #pi/2#.

That is, we have vertical asymptotes when

#(17pi)/12+4x = pi/2 + pik# #" "# (#k# an integer)

#4x = (-11pi)/12+pik#

#4x=(-11pi)/48 + pi/4k#

Note that #tan((-11pi)/12) = tan(pi/12)# so there are vertical asymptotes at

#4x = pi/12 + pi k# #" "# (#k# an integer)

#x=pi/48+pi/4k# with #k# an integer.

(We could have left the solution as #x=(-11pi)/48 + pi/4k#, but I like the positive reference number (angle).)