This problem involves a limit of the form:

$\lim_{h \to 0} \frac{R(5 - h) - R(5)}{5 - h - 5}.$

It resembles the definition of the derivative of a function $R(x)$ at $x = 5$, which is given as:

$R'(5) = \lim_{h \to 0} \frac{R(5 - h) - R(5)}{-h}.$

From the given information, $R(5) = 2$. Let's compute this step by step.

Step 1: Identify the denominator

The denominator $5 - h - 5$ simplifies to $-h$.

Thus, the expression becomes:

$\lim_{h \to 0} \frac{R(5 - h) - R(5)}{-h}.$

Step 2: Apply the derivative definition

This is exactly the definition of $R'(5)$. Therefore, this limit represents $R'(5)$.

Step 3: Conclusion

If additional details about $R(x)$ or $R'(5)$ are provided, we can compute further. However, with the given information, the value of the limit is $R'(5)$.

Do you know $R'(5)$ or want further exploration? 😊

Here are 5 related questions:

1. What is the definition of a derivative, and how does it relate to this limit problem?
2. How would this limit change if $R(x)$ were explicitly defined as a polynomial or trigonometric function?
3. Can this approach be extended to higher-order derivatives (e.g., $R''(5)$)?
4. What are some real-world applications of derivative limits like this one?
5. How can you verify the result of such a limit using numerical approximation?

Tip: Always simplify the denominator before substituting into limits—it often reveals the definition of a derivative.