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Let's evaluate each of the statements to determine which are always true.

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

This statement is **false**. Differentiability implies continuity, but the converse is not necessarily true. A function can fail to be differentiable at a point (for example, having a cusp or a sharp corner) but can still have a well-defined limit at that point. A classic example is \( f(x) = |x| \), which is not differentiable at \( x = 0 \) but still continuous at \( x = 0 \) with \( \lim_{x \to 0} f(x) = 0 \).

### (ii) "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 given limit is the definition of the derivative of \( f \) at \( x = b \). If the derivative exists, the function must be continuous at that point, which implies that \( \lim_{x \to b} f(x) = f(b) \). Differentiability requires continuity.

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

This statement is **false**. Continuity alone does not imply differentiability. A function can be continuous but not differentiable at a point, as in the example \( f(x) = |x| \) at \( x = 0 \). While the function is continuous at \( x = 0 \), it is not differentiable there because the left-hand and right-hand slopes differ.

### Conclusion:
- Statement **(ii)** is always true.
- Statements **(i)** and **(iii)** are false.

Let me know if you'd like further clarification or more details!

### Related Questions:
1. What is the difference between continuity and differentiability?
2. Can you give an example of a function that is continuous but not differentiable at a point?
3. What conditions must be met for a function to be differentiable at a point?
4. How does the existence of a derivative relate to limits in general?
5. Can a function be differentiable at every point but still have discontinuities?

### Tip:
When assessing differentiability, always check for continuity first. If a function is not continuous at a point, it cannot be differentiable there.

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

Explore which statements about limits, continuity, and differentiability are always true. This problem involves concepts like limits, differentiability, and their implications in calculus. Suitable for high school or college-level students.

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