Question

find the value of limit

Answer 1

$lim _{x - > 0} \:( \csc(x) - \cot(x) ) \\ put \: x \: = 0 \: \\ we \: get \: a \: \infty - \infty \: form \: \\ we \: need \: to \: simplify \: it \: ... \\ put \: 1 = \csc^{2} (x) - \cot^{2} (x) in \: the \: denominator \: \\ = lim _{x - > 0} \: \: \frac{\csc(x) - \cot(x) }{\csc^{2} (x) - \cot^{2} (x)} \: \\ = lim _{x - > 0} \: \: \: \frac{1}{( \csc(x) - \cot(x) )} \\ = lim _{x - > 0} \: \: \: \frac{1}{ \frac{1}{ \sin(x) } - \frac{ \cos(x) }{ \sin(x) } } \\ = lim _{x - > 0} \: \: \: \frac{ \sin(x) }{1 - \cos(x) } \\ put \: x \: = 0 \\ = > \: \: \: \frac{ \sin(0) }{1 - \cos(0) } \\ Hence \: \: \: lim _{x - > 0} \: \: = 0$