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.