Step-by-step explanation:
[tex] = \lim \limits_{x \to0} \frac{x - \sin(2x) }{x + \sin(3x) } [/tex]
[tex] = \lim \limits_{x \to0} \frac{ \frac{d}{dx}(x - \sin(2x) ) }{ \frac{d}{dx} (x + \sin(3x) )} [/tex]
[tex] = \lim \limits_{x \to0} \frac{1 - 2\cos(2x) }{1 + 3 \cos(3x) } [/tex]
[tex] = \lim \limits_{x \to0} \frac{ \frac{d}{dx}(1 + 2 \cos(2x) ) }{ \frac{d}{dx} (1 + 3 \cos(3x) )} [/tex]
[tex] = \lim \limits_{x \to0} \frac{ - 4 \sin(2x) }{ - 9 \sin(3x) } [/tex]
[tex] = \frac{ - 4.2}{ - 9.3} [/tex]
[tex] = \frac{ - 8}{ - 27} [/tex]
[tex] = \frac{8}{27} [/tex]
