find lim_(x→0) ((ln(2−cos(2x)))/(ln(1+xsin(3x))))


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Question Number 73330 by mathmax by abdo last updated on 10/Nov/19

$f i n d {l i m}_{x \to 0} \frac{l n (2 - c o s (2 x))}{l n (1 + x s i n (3 x))}$
Commented by mathmax by abdo last updated on 11/Nov/19

$l e t f (x) = \frac{l n (2 - c o s (2 x))}{l n (1 + x s i n (3 x))} w e h a v e c o s (2 x) \sim 1 - \frac{{(2 x)}^{2}}{2} = 1 - 2 x^{2} \Rightarrow$ $2 - c o s (2 x) \sim 2 - 1 + 2 x^{2} = 1 + 2 x^{2} a n d l n (1 + 2 x^{2}) \sim 2 x^{2} (x \sim 0) \Rightarrow$ $a l s o x s i n (3 x) \sim x (3 x) = 3 x^{2} \Rightarrow f (x) \sim \frac{2 x^{2}}{3 x^{2}} \Rightarrow {l i m}_{x \to 0} f (x) = \frac{2}{3}$
	Answered by Smail last updated on 10/Nov/19

	$l n (2 - c o s (2 x)) = l n (1 + 2 {s i n}^{2} (x)) \underset{0}{\sim} 2 {s i n}^{2} (x)$ $l n (1 + x s i n (3 x)) \underset{0}{\sim} x s i n (3 x)$ $\underset{x \to 0}{l i m} \frac{l n (2 - c o s (2 x))}{l n (1 + x s i n (3 x))} = \underset{x \to 0}{l i m} \frac{2 {s i n}^{2} x}{x s i n (3 x)}$ $= \underset{x \to 0}{l i m} 2 \frac{s i n x}{x} \times \frac{s i n x}{x} \times \frac{3 x}{s i n (3 x)} \times \frac{1}{3}$ $= \frac{2}{3}$
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