The problem asks you to calculate $f'(5)$ by finding the limit:

$\lim_{x \to 5} \frac{f(x) - f(5)}{x - 5}$

This expression represents the definition of the derivative of the function $f(x)$ at $x = 5$. To solve this, you'll need to:

1. Determine the function $f(x)$: This is necessary to compute the limit.
2. Evaluate $f(5)$: Find the value of the function at $x = 5$.
3. Compute the limit: Substitute these values into the limit expression and solve.

However, the specific function $f(x)$ isn't provided in the image. If you know what $f(x)$ is, please share that, and I can help you calculate the derivative.

Would you like to proceed with a specific function $f(x)$, or do you have any further details to provide? Here are some related questions that might be helpful:

1. What is the function $f(x)$ in this problem?
2. Can you evaluate $f(5)$ for a given function?
3. How do you apply the definition of the derivative in different scenarios?
4. What methods can be used if the limit is indeterminate?
5. How can the limit definition of the derivative be visualized graphically?

Tip: The limit definition of the derivative is fundamental in calculus, as it gives a precise meaning to the concept of the slope of the tangent line to a curve at a point.