lim_(x→0) ((x^2 +sin3x)/(2x−sinx))=?


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Question Number 171341 by mathlove last updated on 13/Jun/22

$\lim_{x \to 0} \frac{x^{2} + s i n 3 x}{2 x - s i n x} = ?$
Commented by infinityaction last updated on 13/Jun/22

$\lim_{x \to 0} \frac{x (x + \frac{3 \sin 3 x}{3 x})}{x (2 - \frac{\sin x}{x})}$ $\frac{0 + 3}{2 - 1} = 3$
	Answered by Mathspace last updated on 13/Jun/22

	$s i n (3 x) \sim 3 x a n d s i n x \sim x \Rightarrow$ $\frac{x^{2} + 3 x}{2 x - s i n x} \sim \frac{x^{2} + 3 x}{2 x - x} = \frac{x (x + 3)}{x}$ $= x + 3 \Rightarrow$ ${l i m}_{x \to 0} \frac{x^{2} + s i n (3 x)}{2 x - s i n x} = 3$
	Commented by mathlove last updated on 13/Jun/22

	$t h a n k s$
	Answered by nurtani last updated on 13/Jun/22

	$\lim_{x \to 0} \frac{x^{2} + s i n 3 x}{2 x - s i n x} = \lim_{x \to 0} \frac{2 x + 3 c o s 3 x}{2 - c o s x}$ $= \frac{2 (0) + 3 c o s 3 (0)}{2 - c o s 0}$ $= \frac{0 + 3}{2 - 1}$ $= 3$
	Commented by mathlove last updated on 13/Jun/22

	$w i t h o u t H s p i t a l r u l s$
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