lim-x-0-3x-sin2x-2x-sin3x-

Question Number 2265 by B744237509 last updated on 12/Nov/15

\lim_{x \to 0} \frac{3 x + s i n 2 x}{2 x + s i n 3 x} = ?

Answered by Filup last updated on 12/Nov/15

= \lim_{x \to 0} (\frac{3 x}{2 x + \sin (3 x)} + \frac{\sin (2 x)}{2 x + \sin (3 x)})

U s e L^{'} H {\hat{o p i t a l}}^{'} s l a w

= \lim_{x \to 0} (\frac{3}{2 + 3 \cos (3 x)} + \frac{2 \cos (2 x)}{2 + 3 \cos (3 x)})

\cos (0) = 1

= \frac{3}{2 + 3 \cos (3 \times 0)} + \frac{2 \cos (2 \times 0)}{2 + 3 \cos (3 \times 0)}

= \frac{3}{2 + 3} + \frac{2}{2 + 3}

= \frac{3}{5} + \frac{2}{5}

= 1

∴ \lim_{x \to 0} \frac{3 x + \sin (2 x)}{2 x + \sin (3 x)} = 1

Commented by Filup last updated on 12/Nov/15

Expanding the fraction into two parts

isnt nessisary . You can start on line 2

Answered by Rasheed Soomro last updated on 12/Nov/15

A w a y w h i c h {d o e s n}^{'} t u s e d e r i v a t i v e .

\lim_{x \to 0} \frac{3 x + s i n 2 x}{2 x + s i n 3 x}

= \lim_{x \to 0} \frac{\frac{3 x + s i n 2 x}{2 x} \times 2 x}{\frac{2 x + s i n 3 x}{3 x} \times 3 x}

= \frac{2}{3} \underset{x \to 0}{l i m} \frac{\frac{3 x}{2 x} + \frac{s i n 2 x}{2 x}}{\frac{2 x}{3 x} + \frac{s i n 3 x}{3 x}}

= \frac{2}{3} \underset{x \to 0}{l i m} \frac{\frac{3}{2} + \frac{s i n 2 x}{2 x}}{\frac{2}{3} + \frac{s i n 3 x}{3 x}}

= \frac{2}{3} \times \frac{\frac{3}{2} + \underset{2 x \to 0}{l i m} \frac{s i n 2 x}{2 x}}{\frac{2}{3} + \underset{3 x \to 0}{l i m} \frac{s i n 3 x}{3 x}} [x \to 0 {\begin{cases} 2 x \to 0 \\ 3 x \to 0 \end{cases}]

= \frac{2}{3} \times \frac{\frac{3}{2} + 1}{\frac{2}{3} + 1} [\underset{x \to 0}{l i m} \frac{s i n x}{x} = 1]

= \frac{2}{3} \times \frac{5}{2} \times \frac{3}{5} = 1

Commented by Filup last updated on 12/Nov/15

I knew there was a way but i didnt know

how to do it . Nice!

Commented by B744237509 last updated on 12/Nov/15

t h a n x