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Question Number 53376 by rajeshghorai130@gmail.com last updated on 21/Jan/19

Commented by Abdo msup. last updated on 21/Jan/19

$l e t A (x) = \frac{{(s i n α)}^{x} - {(c o s α)}^{x} - c o s (2 α)}{x - 4}$ $A (x) =_{x - 4 = t} \frac{{(s i n α)}^{t + 4} - {(c o s α)}^{t + 4} - c o s (2 α)}{t} = B (t)$ $\Rightarrow {l i m}_{x \to 4} A (x) = {l i m}_{t \to 0} B (t)$ $B (t) = \frac{{s i n}^{4} α e^{t l n (s i n α)} - {(c o s α)}^{4} e^{t l n (c o s α)} - c o s (2 α)}{t}$ $b u t e^{t l n (s i n α)} \sim 1 + t l n (s i n α) (t \to 0)$ $e^{t l n (c o s α)} \sim 1 + t l n (c o s α) \Rightarrow$ $B (t) \sim \frac{{s i n}^{4} α (1 + t l n (s i n α)) - ({c o s}^{4} α) (1 + t l n (c o s α)) - c o s (2 α)}{t}$ $B (t) \sim \frac{{s i n}^{4} α - {c o s}^{4} α - c o s (2 α) + {t s i n}^{4} α l n (s i n α) - {t c o s}^{4} α l n (c o s α)}{t}$ $\Rightarrow B (t) \sim {s i n}^{4} α l n (s i n α) - {c o s}^{4} α l n (c o s α) (t \to 0)$ $\Rightarrow {l i m}_{x \to 4} A (x) = {l i m}_{t \to 0} B (t)$ ${(s i n α)}^{4} l n (s i n α) - {(c o s α)}^{4} l n (c o s α) .$
	Answered by pooja24 last updated on 21/Jan/19

	$u s i n g l^{'} h o p i t a l r u l e a n s i s 4 ({s i n}^{3} α - {c o s}^{3} α)$
	Answered by tanmay.chaudhury50@gmail.com last updated on 21/Jan/19

	$i t h i n k q u e s t i o n s h o u l d b e$ $\lim_{x \to 4} \frac{{(s i n α)}^{x} - {(c o s α)}^{x} + c o s 2 α}{x - 4} (\frac{0}{0}) f o r m$ $u s i n g L H r u l e$ $\lim_{x \to 4} \frac{{(s i n α)}^{x} l n s i n α - {(c o s α)}^{x} l n c o s α}{1}$ $= {(s i n α)}^{4} l n s i n α - {(c o s α)}^{4} l n c o s α$
	Commented by rajeshghorai130@gmail.com last updated on 21/Jan/19

	$solve it without LH rule ? ?$
	Commented by tanmay.chaudhury50@gmail.com last updated on 21/Jan/19

	$i s i t a r e q u e s t . . . l a n g u a g e . . s e e m o t h e r w i s e$
	Commented by tanmay.chaudhury50@gmail.com last updated on 21/Jan/19

	$w i t h o u t L H r u l e . . .$ $t = x - 4$ $\lim_{t \to 0} \frac{{(s i n α)}^{t + 4} - {(c o s α)}^{t + 4} + {c o s}^{2} α - {s i n}^{2} α}{t}$ $\lim_{t \to 0} \frac{{({s i n}^{} α)}^{t + 4} - {(c o s α)}^{t + 4} + ({c o s}^{2} α - {s i n}^{2} α) ({c o s}^{2} α + {s i n}^{2} α) \leftarrow t r i c k s}{t}$ $= \lim_{t \to 0} [\frac{{s i n}^{4} α ({s i n}^{t} α - 1)}{t} - \frac{{c o s}^{4} α ({c o s}^{t} α - 1)}{t}]$ $= {s i n}^{4} α \times l n s i n α - {c o s}^{4} α \times l n c o s α (p r o v e d)$ $u s e d f o r m u l a$ $\lim_{x \to 0} \frac{a^{x} - 1}{x} = l n a$ $N$ $\underset{x \to 0}{li}$ $\lim_{x \to 0} \frac{a^{x} - 1}{x}$ $= \underset{x \to 0}{li}$
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