For what values of x, if any, does #f(x) = 1/((x-5)sin(pi+1/x) # have vertical asymptotes?

1 Answer
Feb 24, 2018

Answer:

#1/((x-5)sin(pi+1/x))# has vertical asymptotes at #x=5# and #x=1/(mpi)#

Explanation:

Vertical asymptotes of #1/((x-5)sin(pi+1/x))# are there when

either #x-5=0# i.e. #x=5#

or when #sin(pi+1/x)=0# i.e. #pi+1/x=npi#, where #n# is an integer

or #1/x=pi(n-1)# or #x=1/(pi(n-1))=1/(mpi)#, where #m# is an integer so that #m=n-1#. Note that as #m# increases, value of #x# decreases continuously and maximum value (less than #5#) is #x=1/pi=0.3183#

graph{1/((x-5)sin(pi+1/x)) [-0.996, 0.996, -0.498, 0.498]}

graph{1/((x-5)sin(pi+1/x)) [-14.41, 17.46, -8.86, 7.08]}