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Question Number 192573 by senestro last updated on 21/May/23

s o l v e;

\lim_{x \to 0} x^{2} t a n (\frac{s i n π x}{2 x})

s o l u t i o n

l e t L = \lim_{x \to 0} x^{2} t a n (\frac{s i n π x}{2 x})

s i n c e s i n x \sim x - \frac{x^{3}}{6}

L = \lim_{x \to 0} x^{2} t a n (\frac{π x}{2 x} - \frac{π^{3} x^{3}}{12 x})

L = \lim_{x \to 0} x^{2} t a n (\frac{π}{2} - \frac{π^{3} x^{2}}{12})

s i n c e t a n (\frac{π}{2} - x) = \frac{1}{t a n x}

L = \lim_{x \to 0} \frac{x^{2}}{t a n (\frac{π^{3} x^{2}}{12})}

L = \lim_{x \to 0} \frac{\frac{π^{3} x^{2}}{12}}{t a n (\frac{π^{3} x^{2}}{12})} \frac{12}{π^{3}}

L = \frac{12}{π^{3}} \lim_{x \to 0} \frac{\frac{π^{3} x^{2}}{12}}{t a n (\frac{π^{3} x^{2}}{12})}

s i n c e \lim_{x \to 0} \frac{x}{t a n x} = 1

L = \frac{12}{π^{3}} \cdot 1 = \frac{12}{π^{3}}

s o l v e d b y H Y a . k . a s e n e s t r o