How do you determine the limit of #cot(x)# as x approaches #pi^-#?

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Answer 1 · 2016-07-01T22:11:22

# = - oo#

#lim_{x to pi^-} cot x#

#= lim_{x to pi^-} cosx/sinx#

let #x = pi - eta#, where #0 < eta < < 1#

so
#= lim_{x to pi^-} cosx/sinx = lim_{eta to 0} cos(pi - eta)/sin(pi - eta)#

# lim_{eta to 0} (cos(pi) cos(eta) + sin(pi) sin(eta))/(sin(pi) cos(eta) - sin(eta)cos(pi))#

# lim_{eta to 0} ((-1) cos(eta) + (0) sin(eta))/((0) cos(eta) - sin(eta)(-1))#

# lim_{eta to 0} - ( cos(eta) )/( sin(eta))#

# - cos(0) lim_{eta to 0} 1/( sin(eta))#

# = - oo#