The function in the image is:

$f(x) = \frac{\sin(\sqrt{x})}{x - 2}$

This function has a square root in the numerator and a difference in the denominator. Here are some key observations:

1. Domain of the function: Since $\sqrt{x}$ requires that $x \geq 0$ (to avoid taking the square root of a negative number), the domain is restricted to $x \geq 0$. Additionally, we must exclude $x = 2$ because it would cause a division by zero. Therefore, the domain of $f(x)$ is $x \in [0, 2) \cup (2, \infty)$.

2. Continuity: The function might be discontinuous at $x = 2$, and we may want to check whether there is a limit as $x \to 2$ from both sides.

3. Derivative: If you'd like to find the derivative, we would need to apply the quotient rule along with the chain rule for the sine and square root functions.

4. Behavior at $x = 2$: The limit as $x \to 2$ could be of interest, especially in terms of whether the function exhibits a removable discontinuity or a vertical asymptote.

Would you like a detailed exploration of the domain, limits, or differentiation of this function? Feel free to ask!

Questions to Expand on This:

1. How do we determine the limit of this function as $x$ approaches 2?
2. Can the discontinuity at $x = 2$ be removed?
3. What is the derivative of this function using the quotient rule?
4. How do we interpret the behavior of the sine function for small values of $\sqrt{x}$?
5. What role does the chain rule play in differentiating the numerator?

Tip:

When analyzing functions with square roots and trigonometric terms, always first consider the domain restrictions caused by both components.