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Question Number 66379 by Sandy Suhendra last updated on 13/Aug/19

Commented by kaivan.ahmadi last updated on 13/Aug/19

$= {l i m}_{x \to \frac{π}{2}} \frac{3 c o s x + 5 c o s 3 x}{c o t 5 x} \overset{h o p}{=}$ ${l i m}_{x \to \frac{. π}{2}} \frac{- 3 s i n x - 15 s i n 3 x}{- 5 (1 + {c o t}^{2} 5 x)} = \frac{- 3 + 15}{- 5} = - \frac{12}{5}$
Commented by mathmax by abdo last updated on 14/Aug/19

$l e t A (x) = (3 c o s x + 5 c o s (3 x)) t a n (5 x) c h a n g e m e n t t = \frac{π}{2} - x g i v e$ ${l i m}_{x \to \frac{π}{2}} A (x) = {l i m}_{t \to 0} (3 s i n t + 5 c o s (\frac{3 π}{2} - 3 t)) t a n (\frac{5 π}{2} - 5 t)$ $= {l i m}_{t \to 0} (3 s i n t - 5 s i n (3 t)) \times \frac{1}{t a n (5 t)} w e h a v e$ $s i n t \sim t, s i n (3 t) \sim 3 t a n d t a n (5 t) \sim 5 t \Rightarrow$ $\frac{3 s i n t - 5 s i n (3 t)}{t a n (5 t)} \sim \frac{3 t - 15 t}{5 t} = - \frac{12}{5} \Rightarrow {l i m}_{x \to \frac{π}{2}} A (x) = - \frac{12}{5}$
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