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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**

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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.

Which of the following statements are always true?
(i) If lim_{x→b} f(x) = f(b) then f is differentiable at x = b.
(ii) If f is not differentiable at x = b, then lim_{x→b} f(x) does not exist.
(iii) If lim_{x→b} (f(x) - f(b)) / (x - b) exists, then lim_{x→b} f(x) = f(b).

Learn which mathematical statements involving limits, continuity, and differentiability are always true. Explore key concepts such as the definition of a derivative, the relationship between continuity and differentiability, and how limits work at a point.

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