Let $f(x) = x^{1/3}$.

1. Use the formal definition of continuity to determine if $f(x)$ is continuous at $x = 0$.

   Hint (a):

   

2. Calculate $\lim\limits_ { x \rightarrow 0 } \frac { f ( x ) - f ( 0 ) } { x - 0 }$.

   Hint (b):

   This is Ana's definition of the derivative. The limit is equivalent to $f^\prime(0)$.So you can use the power rule to find the limit, if it exists.

3. What does the limit in part (b) represent?

   Hint (c):

   Refer to the hint in part (b).

4. What is happening graphically on $f(x)$ at $x = 0$ that causes your answer to part(b)?

   Hint 1(d):

   Recall that derivatives do not exist at locations where $f(x)$ has a: cusp, endpoint, jump, hole or vertical tangent.

   Hint 2(d):

   Sketching the graph may help visualize the issue.