Mathematics

This limit tests whether a rapidly decaying exponential can "kill off" an oscillating function. The key insight is that $\sin x$ is bounded—it can never exceed 1 in absolute value—so even though it wiggles forever, the exponential $e^{-x}$ shrinks to zero fast enough to drag the entire product down. Use the squeeze theorem by bounding $|\sin x|$ and watching what happens as $x$ grows large.

This limit tests whether a rapidly decaying exponential can "kill off" an oscillating function. The key insight is that $\sin x$ is bounded—it can never exceed 1 in absolute value—so even though it wiggles forever, the exponential $e^{-x}$ shrinks to zero fast enough to drag the entire product down. Use the squeeze theorem by bounding $|\sin x|$ and watching what happens as $x$ grows large.