(1) \( \lim _{x \rightarrow 0} \frac{x}{a}\left[\frac{b}{x}\right] \)

Let's dive into this limit! As \( x \) approaches 0, the term \( \frac{b}{x} \) grows infinitely large if \( b \neq 0 \). Thus, the whole expression transforms into \( \lim _{x \rightarrow 0} \frac{x}{a} \cdot \frac{b}{x} = \lim _{x \rightarrow 0} \frac{b}{a} \), which does not depend on \( x \). Therefore, the limit evaluates to \( \frac{b}{a} \). Now, a key point to remember here is that \( a \) should not be zero; otherwise, the limit would be undefined due to division by zero. If we assume \( a \neq 0 \), we've turned a potentially tricky limit into a straightforward expression. So, always keep an eye on assumptions when dealing with such limits!