Chapter 5 – Continuity and Differentiability

1. State the condition for continuity of f(x) at x=a.

2. Is every differentiable function continuous?

3. Evaluate lim x→0 (sin x)/x.

4. Find d/dx(x³).

5. Is |x| differentiable at x=0?

6. State the derivative of eˣ.

1. Test the continuity of f(x)=x² at x=2.

2. Differentiate x sin x using the product rule.

3. Find the derivative of log(sin x).

1. Find k so that f(x)=(x²−4)/(x−2) for x≠2 and f(2)=k is continuous at x=2.

2. Using logarithmic differentiation, differentiate xˣ.

3. Prove that a differentiable function must be continuous.

1. Investigate the continuity and differentiability of f(x)=|x−1| at x=1. Explain why continuity does not imply differentiability.