Question

$\underset{x\to \infty }{lim}xcos\left(\pi /4x\right)sin\left(\pi /4x\right)$ is equal to -

Answer 1

$\underset{x\to \infty }{lim}\frac{x}{2}\left[2cos\right(\frac{\pi }{4x}\left)sin\right(\frac{\pi }{4x}\left)\right]$

$=\underset{x\to \infty }{lim}\frac{x}{2}\left(2cos\right(\frac{\pi }{4x}\left)sin\right(\frac{\pi }{4x}\left)\right)$

$=\underset{x\to \infty }{lim}\frac{x}{2}sin\left(\frac{\pi }{2x}\right)$

$=\underset{x\to \infty }{lim}\frac{sin\left(\pi /2x\right)}{\pi /2x}\times \frac{\pi }{4}=\frac{\pi }{4}$............[$\underset{y\to 0}{lim}\frac{siny}{y}=1$ where $y=\pi /2x$]