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

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

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

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

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

$= - oo$

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