4. HaǎTu mpertest: lim _(xarrow infty )(sin3x)/(sin5x)

To evaluate the limit \(\lim_{x \to \infty} \frac{\sin(3x)}{\sin(5x)}\), we can use the fact that \(\sin(kx)\) for any constant oscillates between -1 and 1 as increases. Specifically, \(\sin(kx)\) does not settle down to a single value but continues to oscillate.To see this more formally, consider the behavior of the sine functions: Both \(\sin(3x)\) and \(\sin(5x)\) oscillate between -1 and 1 as increases. Since the sine function oscillates indefinitely, the ratio \(\frac{\sin(3x)}{\sin(5x)}\) does not converge to a single finite value. Instead, it continues to oscillate.To make this more precise, let's consider the period of the sine functions. The period of \(\sin(3x)\) is , and the period of \(\sin(5x)\) is . As approaches infinity, both \(\sin(3x)\) and \(\sin(5x)\) will continue to oscillate without settling down.Given this, we can conclude that: does not exist because the ratio continues to oscillate indefinitely. Therefore, the limit is undefined.So, the final answer is: