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Question
Evaluate `Lim _(x→0) (cot x)^sinx.`
Solution
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`
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