The expression $-\frac{\sin(x)}{x}$ involves a trigonometric function and a rational term. Here's an analysis:

Behavior of $-\frac{\sin(x)}{x}$

1. For $x = 0$:
   This expression is undefined because division by zero is not allowed. However, the limit of $-\frac{\sin(x)}{x}$ as $x$ approaches 0 can be evaluated using L'Hôpital's Rule: $\lim_{x \to 0} -\frac{\sin(x)}{x} = \lim_{x \to 0} -\frac{\cos(x)}{1} = -1.$

2. For $x > 0$:
   The sine function oscillates between $-1$ and $1$, and dividing by $x$ (positive values) causes the magnitude of the function to decrease as $x$ increases. The function approaches 0 from below as $x \to \infty$.

3. For $x < 0$:
   Similar behavior occurs, except $x$ is negative, making the denominator negative. Thus, the function still approaches 0, but from above, as $x \to -\infty$.

4. Overall Characteristics:

   • The function is odd, meaning $f(-x) = -f(x)$.
   • There is a singularity at $x = 0$.
   • The function tends to 0 as $x \to \pm\infty$.

Would you like to:

1. Evaluate the function for specific values?
2. Explore its graph or behavior further?
3. Derive its limit at infinity or near zero?
4. Find the derivative or integral of this function?
5. Analyze any approximations or expansions, such as Taylor series?

Tip: When analyzing functions with a singularity at $x = 0$, using limits is a powerful tool to understand the behavior.