lim_(x→0) ((sin5x − tan5x)/x^3 )


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Question Number 106825 by 175mohamed last updated on 07/Aug/20

$\lim_{x \to 0} \frac{s i n 5 x - t a n 5 x}{x^{3}}$
	Answered by bemath last updated on 07/Aug/20

	$^{@ bemath @}$ $\lim_{x \to 0} \frac{\tan 5 x (\cos 5 x - 1)}{x^{3}} =$ $\lim_{x \to 0} \frac{\tan 5 x (- 2 \sin^{2} (\frac{5 x}{2}))}{x^{3}} = - 2 \times 5 \times \frac{25}{4}$ $= - \frac{125}{2}$
	Answered by Dwaipayan Shikari last updated on 07/Aug/20

	$\lim_{x \to 0} \frac{s i n 5 x (c o s 5 x - 1)}{c o s 5 x . x^{3}} = \lim_{x \to 0} \frac{- 5 x . 2 {s i n}^{2} (\frac{5 x}{2})}{x^{3}} = \lim_{x \to 0} \frac{- 5 x . 2 {(\frac{5 x}{2})}^{2}}{x^{3}} = - \frac{125}{2}$
	Answered by Dwaipayan Shikari last updated on 07/Aug/20

	$A n o t h e r w a y$ $\lim_{x \to 0} \frac{5 x - \frac{{(5 x)}^{3}}{6} - 5 x - \frac{{(5 x)}^{3}}{3}}{x^{3}} = \lim_{x \to 0} \frac{- (\frac{375 x^{3}}{6})}{x^{3}} = - \frac{125}{2}$
	Answered by malwaan last updated on 07/Aug/20

	$\underset{x \to 0}{l i m} \frac{s i n 5 x - \frac{s i n 5 x}{c o s 5 x}}{x^{3}}$ $= \underset{x \to 0}{l i m} \frac{s i n 5 x c o s 5 x - s i n 5 x}{x^{3} c o s 5 x}$ $= \underset{x \to 0}{l i m} \frac{s i n 5 x [c o s 2 (\frac{5 x}{2}) - 1]}{x^{3} c o s 5 x}$ $= \underset{x \to 0}{l i m} \frac{s i n 5 x [- 2 {s i n}^{2} (\frac{5 x}{2})]}{x^{3} c o s 5 x}$ $= \frac{5 [- 2 {(\frac{5}{2})}^{2}]}{1^{3} \times 1} = - \frac{125}{2}$
	Commented by JDamian last updated on 07/Aug/20

	$W h y x^{3} \Rightarrow 1^{3} ?$
	Commented by malwaan last updated on 07/Aug/20

	$s i r J D a m i a n$ $i t i s s h o r t c u t m e t h o d w i t h$ $s i n x / t a n x / x$
	Answered by 1549442205PVT last updated on 07/Aug/20

	$\lim_{x \to 0} \frac{s i n 5 x - t a n 5 x}{x^{3}} = \underset{L^{'} Hopital}{} \lim_{x \to 0} \frac{5 \cos 5 x - (1 + \tan^{2} 5 x) . 5}{{3 x}^{2}}$ $\underset{L^{'} Hopital}{=} \lim_{x \to 0} \frac{- 25 \sin 5 x - 5 . [2 \tan 5 x (1 + \tan^{2} 5 x) . 5]}{6 x}$ $= \lim_{x \to 0} \frac{- 25 \sin 5 x - 50 \tan 5 x - {50 \tan}^{3} 5 x}{6 x}$ $= l \underset{x \to 0}{im} (\frac{- 125}{6} \times \frac{\sin 5 x}{5 x}) - \lim_{x \to 0} (\frac{250}{6 \cos 5 x} \times \frac{\sin 5 x}{5 x})$ $- \lim_{x \to 0} (\frac{250}{6 cosx} \times \frac{\sin 5 x}{5 x} \times \tan^{2} x)$ $= (\frac{- 125}{6} \times 1) - (\frac{250}{6} \times 1) - (\frac{250}{6} \times 1 \times 0)$ $= \frac{- 375}{6} = \frac{- 125}{2}$
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