Question

Evaluate: $\underset{x\to \infty }{lim}x\sin \frac{1}{x}$

Answer 1

${lim}_{x\to \infty }x\sin \frac{1}{x}$

$={lim}_{x\to \infty }\frac{x\sin \frac{1}{x}}{\frac{1}{x}\times x}$

$={lim}_{x\to \infty }\frac{\sin \frac{1}{x}}{\frac{1}{x}}$

$=1$ since ${lim}_{\theta \to 0}\frac{\sin \theta }{\theta }=1$