find without using l′hopital lim_(x→0) ((ln(1+cos(x))−ln(2))/x)


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Question Number 82887 by M±th+et£s last updated on 25/Feb/20

$f i n d w i t h o u t u s i n g l^{'} h o p i t a l$ $\underset{x \to 0}{l i m} \frac{l n (1 + c o s (x)) - l n (2)}{x}$
Commented by msup trace by abdo last updated on 25/Feb/20

$w e h a v e c o s x \sim 1 - \frac{x^{2}}{2} \Rightarrow$ $2 + c o s x \sim 2 - \frac{x^{2}}{2} \Rightarrow$ $l n (1 + c o s x) \sim l n (2) + l n (1 - \frac{x^{2}}{4})$ $\sim l n (2) - \frac{x^{2}}{4} \Rightarrow \frac{l n (1 + c o s x) - l n 2}{x}$ $\sim - \frac{x}{4} \to 0 (x \to 0) \Rightarrow$ ${l i m}_{x \to 0} \frac{l n (1 + c o s x) - l n 2}{x} = 0$
	Answered by TANMAY PANACEA last updated on 25/Feb/20

	$\lim_{x \to 0} \frac{l n (\frac{1 + c o s x}{2})}{x}$ $\lim_{x \to 0} \frac{l n (1 + \frac{1 + c o s x}{2} - 1)}{(\frac{1 + c o s x}{2} - 1)} \times \frac{\frac{1 + c o s x}{2} - 1}{x}$ $t = (\frac{1 + c o s x}{2} - 1) w h e n x \to 0 t \to 0$ $\lim_{t \to 0} \frac{l n (1 + t)}{t} \times \lim_{x \to 0} \frac{- 2 s i n \frac{x}{2} \times s i n \frac{x}{2}}{\frac{x}{2} \times 2}$ $1 \times - 1 \times 0 = 0$ $t$ $\lim_{x \to 0}$
	Commented by M±th+et£s last updated on 25/Feb/20

	$t h a n k y o u s i r$
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