Answer 1

Let` L= Lim _(x→0) (cot x)^sinx` 

∴` logl=log{lim_(x→0)(cot x)^sinx}`

=`lim_(x→0){log(cotx)^sinx}`

=`lim_(x→0)sinx.log(cot x)`

=`lim_(x→0) log(cot x)/(cosec x)`     `(∞/∞)`

=`lim_(x→0) (1/(cotx) .-cosec^2x)/(-cosec x cot x)`    (L’ Hospital’s Rule) 

=` Lim_x→0 tanx. 1/sin x. tan x`

=` Lim_x→0 tanx . 1/sin x. sin x/cos x`

= `tan o xx 1/cos 0`

∴ `log L = 0 ` 

∴ `L= e^0`

∴` Lim_x→0  (cot x)^sin x=1`