Let's break down each statement from the question to determine which are true:

(i) If $\lim_{x \to b} f(x) = f(b)$, then $f$ is differentiable at $x = b$.

• False. The condition $\lim_{x \to b} f(x) = f(b)$ only tells us that the function is continuous at $x = b$, but not necessarily differentiable. Differentiability requires not only continuity but also that the derivative exists, which may not be the case. For example, the absolute value function $f(x) = |x|$ is continuous at $x = 0$, but not differentiable at $x = 0$.

(ii) If $f$ is not differentiable at $x = b$, then $\lim_{x \to b} f(x)$ does not exist.

• False. A function can be non-differentiable at a point and still have a well-defined limit. For example, again consider $f(x) = |x|$. This function is continuous (and thus has a limit at $x = 0$) but is not differentiable there. Differentiability is a stronger condition than the existence of a limit.

(iii) If $\lim_{x \to b} \frac{f(x) - f(b)}{x - b}$ exists, then $\lim_{x \to b} f(x) = f(b)$.

• True. This is the definition of the derivative. If the derivative exists at $x = b$, then the function must be continuous at $x = b$, which means $\lim_{x \to b} f(x) = f(b)$.

Answer:

Based on the analysis above:

• (i) is false.
• (ii) is false.
• (iii) is true.

Thus, the correct choice is (C): (iii) only.

Would you like a further breakdown of any concept?

Here are some additional related questions:

1. What is the formal definition of a limit?
2. How does continuity relate to differentiability?
3. Can a function be continuous but not differentiable? Provide an example.
4. How do you interpret the limit definition of a derivative?
5. What are examples of functions that are non-differentiable but still continuous?

Tip: Always remember, differentiability implies continuity, but the reverse is not always true!