Question

Question #d7c72

Answer 1

0

Explanation:

`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`