Question #d7c72

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Answer 1 · 2016-08-01T23:16:11

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$csc(x) = 1/(sin(x))$

$cot(x) = 1/(tan(x)) = (cos(x))/(sin(x))$

$1/(sin(x)) - (cos(x))/(sin(x)) = (1-cos(x))/(sin(x))$

$lim_(xrarr0)(1-cos(x))/(sin(x)) = ?$

For L'hopital's rule to be available, need limit to be of indeterminate form.

$(1-cos(0))/(sin(0)) = (1-1)/0 = 0/0$

Hence condition is satisfied. We apply L'hopital's :

$lim_(xrarr0)(d/(dx)(1-cos(x)))/(d/(dx)(sin(x))) = lim_(xrarr0) (sin(x))/(cos(x)) = (sin(0))/(cos(0)) = 0/1 = 0$