lim_(x→0^+ ) (((sin x+(√(1−cos 3x)))/(tan (x(√2)))))= lim_(x→0^− ) (((sin x+(√(1−cos 3x)))/(tan (x(√2)))))= lim_(x→0) (((sin x+(√(1−cos 3x)))/(tan (x(√2)))))=


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Question Number 124010 by john_santu last updated on 30/Nov/20

$\lim_{x \to 0^{+}} (\frac{\sin x + \sqrt{1 - \cos 3 x}}{\tan (x \sqrt{2})}) =$ $\lim_{x \to 0^{-}} (\frac{\sin x + \sqrt{1 - \cos 3 x}}{\tan (x \sqrt{2})}) =$ $\lim_{x \to 0} (\frac{\sin x + \sqrt{1 - \cos 3 x}}{\tan (x \sqrt{2})}) =$
	Answered by liberty last updated on 30/Nov/20
	$(√(1−cos 3x)) = (√(1−(1−2sin^2 (((3x)/2)))) = (√(2sin^2 (((3x)/2)))) = (√2) ∣sin (((3x)/2))∣ = → { (((√2) sin (((3x)/2)) ; x≥0)),((−(√2) sin (((3x)/2)) ; x<0)) :} (1) lim_(x→0^+ ) (((sin x+(√2) sin (((3x)/2)))/(tan (x(√2)))))= ((1+((3(√2))/2))/( (√2))) = (((√2)+3)/2) (2) lim_(x→0^− ) (((sin x−(√2) sin (((3x)/2)))/(tan (x(√2)))))=(((√2)−3)/2) (3) lim_(x→0) (((sin x+(√(1−cos 3x)))/(tan (x(√2))))) doesn′t exist$
	$\sqrt{1 - \cos 3 x} = \sqrt{1 - (1 - 2 \sin^{2} (\frac{3 x}{2})} = \sqrt{2 \sin^{2} (\frac{3 x}{2})}$ $= \sqrt{2} ∣ \sin (\frac{3 x}{2}) ∣ = \to {\begin{cases} \sqrt{2} \sin (\frac{3 x}{2}); x ⩾ 0 \\ - \sqrt{2} \sin (\frac{3 x}{2}); x < 0 \end{cases}$ $(1) \lim_{x \to 0^{+}} (\frac{\sin x + \sqrt{2} \sin (\frac{3 x}{2})}{\tan (x \sqrt{2})}) = \frac{1 + \frac{3 \sqrt{2}}{2}}{\sqrt{2}} = \frac{\sqrt{2} + 3}{2}$ $(2) \lim_{x \to 0^{-}} (\frac{\sin x - \sqrt{2} \sin (\frac{3 x}{2})}{\tan (x \sqrt{2})}) = \frac{\sqrt{2} - 3}{2}$ $(3) \lim_{x \to 0} (\frac{\sin x + \sqrt{1 - \cos 3 x}}{\tan (x \sqrt{2})}) {d o e s n}^{'} t e x i s t$
	Commented by Mammadli last updated on 30/Nov/20

	Commented by john_santu last updated on 30/Nov/20

	$c o n s i d e r a^{3} + b^{3} = (a + b) (a^{2} + b^{2} - a b)$ $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} = \frac{4}{\sqrt[3]{{(2 + \sqrt{5})}^{2}} + \sqrt[3]{{(2 - \sqrt{5})}^{2}} + 1}$
	Commented by malwan last updated on 30/Nov/20

	$= 1$
	Commented by bharathkumar last updated on 30/Nov/20

	$s e x$
	Commented by Dwaipayan Shikari last updated on 30/Nov/20

	$\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} = p$ $2 + \sqrt{5} + 2 - \sqrt{5} + 3 \sqrt{4 - 5} (\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}) = p^{3}$ $4 - 3 p = p^{3}$ $p = 1$
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