To solve question 16 in the provided image:

We are tasked to analyze the function $f(x)$, given that it is differentiable at $x = a$, with $a \neq 0$, and evaluate the limit:

$I = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}.$

This is the definition of the derivative of $f(x)$ at $x = a$. By definition:

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}.$

Hence, the correct statement is:

D: $I = f'(a)$.

Would you like a detailed explanation of the derivative concept or more examples?

Related Questions:

1. What are the key differences between one-sided limits and derivatives?
2. How does the definition of a derivative extend to higher dimensions?
3. Can a function be continuous but not differentiable? Provide an example.
4. How do you apply the derivative to real-world problems involving rates of change?
5. What is the geometric interpretation of a derivative at a point?

Tip:

When working with derivatives, always check for continuity of the function at the given point. Discontinuities can make derivatives undefined.