Question

`\lim_(x \to 0 ) (x-sin(2x))/(x+sin(3x))`

Answer 1

Explanation:

`= \lim \limits_(x \to0) (x - \sin(2x) )/(x + \sin(3x) )`

`= \lim \limits_(x \to0) ( (d)/(dx)(x - \sin(2x) ) )/( (d)/(dx) (x + \sin(3x) ))`

`= \lim \limits_(x \to0) (1 - 2\cos(2x) )/(1 + 3 \cos(3x) )`

`= \lim \limits_(x \to0) ( (d)/(dx)(1 + 2 \cos(2x) ) )/( (d)/(dx) (1 + 3 \cos(3x) ))`

`= \lim \limits_(x \to0) ( - 4 \sin(2x) )/( - 9 \sin(3x) )`

`= ( - 4.2)/( - 9.3)`

`= ( - 8)/( - 27)`

`= (8)/(27)`