9) \( \frac{\log x}{\sin x} \)

Let's explore this intriguing function! The expression \( \frac{\log x}{\sin x} \) combines two fundamental mathematical concepts: logarithms and trigonometric functions. As \( x \) approaches 0 from the positive side, the logarithmic term trends towards negative infinity, while \( \sin x \) approaches 0. This leads us into L'Hôpital's rule territory, where we differentiate the top and bottom to understand the limit's behavior! When considering applications, this function might pop up in fields like signal processing or control theory, where logarithmic scale and oscillatory behavior frequently intersect. For example, when analyzing waves or signals, understanding how these mathematical expressions behave helps in designing systems for optimal response, ensuring smooth signal transitions or minimizing oscillations. It's all about making theoretical math work for real-world scenarios!