To solve this problem, we need to find the interval of continuity for the function $f(x) = \frac{\sqrt{x + 2}}{\sqrt{5 - x}}$.

The conditions for continuity require that:

1. The function is defined (the denominator should not be zero).
2. The function should be within the domain of the square root (the radicands must be non-negative).

Step 1: Denominator (Square root in the denominator)

The denominator contains $\sqrt{5 - x}$, which means: $5 - x > 0 \quad \Rightarrow \quad x < 5$ So, $x$ must be less than 5 to avoid the square root of a negative number and to prevent division by zero (since $\sqrt{0}$ would make the denominator zero).

Step 2: Numerator (Square root in the numerator)

The numerator contains $\sqrt{x + 2}$, so: $x + 2 \geq 0 \quad \Rightarrow \quad x \geq -2$ Thus, $x$ must be greater than or equal to $-2$.

Combining both conditions:

• From the numerator: $x \geq -2$
• From the denominator: $x < 5$

Therefore, the domain of the function is $[-2, 5)$.

Conclusion:

The function is continuous on the interval $[-2, 5)$, which corresponds to option a).

Would you like more details on this solution? Here are five questions to consider next:

1. How do we handle square roots in other types of problems?
2. What happens to a function's continuity if the denominator is negative?
3. Can you graph this function to visualize the intervals?
4. How does continuity affect differentiability?
5. What happens if we modify the numerator or denominator in this function?

Tip: Always check both the numerator and denominator when dealing with rational functions.