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

asked Apr 23, 2022 75.5k views

3 votes

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

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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)

Originalhat answered Apr 27, 2022

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