Let’s analyze question 16 from the uploaded image.

Problem Restatement:

Given a function $f(x)$ that is differentiable at $x = a$, and $a \neq 0$, define $I = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$. The task is to determine the correct statement about $I$.

Solution:

The expression $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ represents the definition of the derivative of $f(x)$ at $x = a$. This is because the derivative of a function is defined as: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}.$

Thus, $I = f'(a)$.

Now we evaluate the provided options:

• A. Không tồn tại giới hạn (The limit does not exist):
  Incorrect. Since $f(x)$ is differentiable at $x = a$, the limit exists by definition.

• B. $I = df(a)$:
  Incorrect. While this expression involves the derivative, it is improperly written. The derivative notation $df(a)$ is not standard.

• C. $I = \frac{df}{dx}(0)$:
  Incorrect. While the derivative notation is valid, the condition says $x = a$, not $x = 0$.

• D. $I = f'(a)$:
  Correct. By definition, the derivative at $x = a$ is represented as $f'(a)$.

Final Answer:

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

Would you like me to elaborate on derivatives or limits? Let me know!

Here are 5 related questions for deeper understanding:

1. What is the definition of differentiability at a point?
2. How does the limit definition of a derivative differ for left-hand and right-hand limits?
3. What are some practical interpretations of derivatives in real-world problems?
4. Can a function be continuous but not differentiable?
5. How is the derivative computed for non-standard functions (e.g., piecewise-defined functions)?

Tip: Always ensure the function is differentiable at the point before applying the derivative definition!