If L=lim_{x rightarrow 0} ((tan x)/x)^{1/x^2}

Emmalee Whitaker

Emmalee Whitaker

Answered question

2023-03-08

If L = lim x 0 ( tan x x ) 1 / x 2 , then the value of 1 ln L is ?

Answer & Explanation

svikaoeol

svikaoeol

Beginner2023-03-09Added 7 answers

L = lim x 0 ( tan x x ) 1 / x 2 [ 1 form]
Next, taking log on both sides, we get
ln L = lim x 0 ln ( tan x x ) x 2 l n L = lim x 0 ln tan x ln x x 2
Using L'Hospital's Rule,
Differntiate with respect to 'x' we get
ln L = lim x 0 sec 2 x tan x 1 x 2 x ln L = lim x 0 x sin x cos x 2 x 2 sin x cos x ln L = lim x 0 2 x sin 2 x 4 x 3
Using L'Hospital's Rule,
Differntiate with respect to 'x' we get
ln L = lim x 0 2 2 cos 2 x 12 x 2 ln L = lim x 0 2 sin 2 x 6 x 2 ln = 2 6 lim x 0 sin 2 x x 2 ln L = 1 3 ( 1 ) 1 ln L = 3

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