find-without-using-l-hopital-lim-x-0-ln-1-cos-x-ln-2-x-

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

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