The problem asks which of the following mathematical statements are always true. Let's analyze each one:

Statement (i):

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

This statement is false. The condition $\lim_{x \to b} f(x) = f(b)$ only ensures that the function is continuous at $x = b$, but continuity does not imply differentiability. A common counterexample is the absolute value function $f(x) = |x|$, which is continuous at $x = 0$ but not differentiable there.

Statement (ii):

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

This statement is also false. Differentiability is a stronger condition than continuity. A function can be continuous (and hence the limit exists) without being differentiable. For example, $f(x) = |x|$ is continuous at $x = 0$, so $\lim_{x \to 0} f(x)$ exists, but $f$ is not differentiable at $x = 0$.

Statement (iii):

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

This statement is true. The expression $\lim_{x \to b} \frac{f(x) - f(b)}{x - b}$ represents the definition of the derivative of $f$ at $x = b$. If the derivative exists at that point, then the function must be continuous at $x = b$, meaning $\lim_{x \to b} f(x) = f(b)$.

Conclusion:

Only statement (iii) is always true. Therefore, the correct answer is:

(C) (iii) only

Would you like more details or have any questions about this explanation?

Here are 5 related questions for further practice:

1. What is the relationship between continuity and differentiability?
2. Can you provide an example of a function that is continuous but not differentiable?
3. Why does the existence of a derivative imply continuity?
4. What is the geometrical interpretation of differentiability at a point?
5. How does the concept of a limit help in determining the differentiability of a function?

Tip: Always remember that differentiability implies continuity, but continuity does not imply differentiability.