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Question Number 203374 by otchereabdullai@gmail.com last updated on 17/Jan/24

	Answered by Calculusboy last updated on 18/Jan/24

	$S o l u t i o n : l i \underset{x \to 0}{m} \frac{t a n 5 x}{3 x} (b y u s i n g a l g e b r a i c m e t h o d s)$ $l i \underset{x \to 0}{m} \frac{t a n 5 x}{3 x} \times \frac{5 x}{5 x} = \frac{5 x}{3 x} \times l i \underset{x \to 0}{m} \frac{t a n 5 x}{5 x}$ $N B : l i \underset{x \to 0}{m} \frac{t a n a x}{x} = 1 t h e n l i \underset{x \to 0}{m} \frac{t a n 5 x}{5 x} = 1$ $∴ l i \underset{x \to 0}{m} \frac{t a n 5 x}{3 x} = \frac{5}{3} \times 1 = \frac{5}{3}$
	Answered by Calculusboy last updated on 18/Jan/24

	$S o l u t i o n : b y u s i n g B F m e t h o d s$ $l e t u = x u^{'} = 1 u^{″} = 0 & v = s i n x v_{1} = - c o s x v_{2} = - s i n x v_{3} = c o s x$ $I = {u v}_{1} - u^{'} v_{2} + u^{″} v_{3} - \cdot \cdot \cdot$ $I = - ∣ x c o s x ∣_{0}^{π} + ∣ s i n x ∣_{0}^{π} + C$ $I = - [π c o s (π) - 0 \cdot c o s (0) ∣ + [s i n (π) - s i n (0)]$ $I = - (- π - 0) + (0 - 0)$ $I = π$
	Answered by Calculusboy last updated on 18/Jan/24

	$S o l u t i o n : (5 B)$ $a) l i \underset{x \to 1}{m} \frac{1 - x}{1 + x} = \frac{1 - 1}{1 + 1} = \frac{0}{2} = 0$ $(b y s u b d i r e c t l y, L^{'} h o s p i t a l r u l e i s d i s c a r d e d)$ $b) l i \underset{x \to 1}{m} \frac{1 + c o s x π}{1 - x} = \frac{1 + c o s π}{1 - 1} = \frac{1 - 1}{1 - 1} = \frac{0}{0}$ $(i n d e t e r m i n a n t, b y u s i n g L^{'} h o s p i t a l r u l e)$
	Commented by otchereabdullai@gmail.com last updated on 18/Jan/24

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