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Evaluate Lim X → 0 ( Cot X ) Sin X . - Applied Mathematics 1

Evaluate `Lim _(x→0) (cot x)^sinx.`

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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`

Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
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