Question

The value of $\underset{x \rightarrow \pi }{l i m}\frac{s i n \left(2 \pi c o s^{2} x\right)}{t a n \left(\pi s e c^{2} x\right)}$ is equal to

• A. $1$
• B. $2$
• C. $-2$
• D. $0$

Answer 1

Answer: (C)

$\underset{x \rightarrow \pi }{l i m}\frac{s i n \left\{2 \pi \left(1 - s i n^{2} x\right)\right\}}{t a n \left\{\pi \left(1 + t a n^{2} x\right)\right\}}=\underset{x \rightarrow \pi }{l i m}\frac{s i n \left\{2 \pi - 2 \pi s i n^{2} x\right\}}{t a n \left\{\pi + \pi t a n^{2} x\right\}}$
$=\underset{x \rightarrow \pi }{l i m}\frac{- s i n \left\{2 \pi s i n^{2} x\right\}}{t a n \left\{\pi t a n^{2} x\right\}}$
$=\underset{x \rightarrow \pi }{lim}\frac{- sin \left\{2 \pi \left(sin\right)^{2} x\right\}}{\left\{2 \pi \left(sin\right)^{2} x\right\}}\times \frac{2 \pi \left(sin\right)^{2} x}{\pi \left(tan\right)^{2} x}\times \frac{\pi \left(tan\right)^{2} x}{tan \left(\pi \left(tan\right)^{2} x\right)}$
$=-1\times 2\times 1=-2$