https://file.aiquickdraw.com/mathbot/file/1722454116895llc9fyrh.jpeg

Here is the graph of the function \( f(x) \) defined by:

\[
f(x) = \begin{cases} 
x^3 & \text{if } x \neq 1 \\
0 & \text{if } x = 1 
\end{cases}
\]

- The blue curve represents \( x^3 \) for all \( x \neq 1 \).
- The red dot represents the point \( (1, 0) \), which indicates that \( f(1) = 0 \).

In summary:
- The left-hand and right-hand limits as \( x \to 1 \) are both 1.
- The limit \( \lim_{x \to 1} f(x) = 1 \), even though \( f(1) = 0 \), demonstrating a discontinuity at \( x = 1 \).

If you have any questions or need further details, feel free to ask!

### Further Questions:
1. How do you determine the continuity of a piecewise function at a point?
2. What are the properties of the cubic function \( x^3 \)?
3. Why is there a discontinuity at \( x = 1 \) in this function?
4. How would the graph change if \( f(1) = 1 \) instead?
5. What are the implications of a function having different values from its limit at a point?
6. How can you use the epsilon-delta definition to prove the limit of \( f(x) \) as \( x \to 1 \)?
7. Can you explain the concept of left-hand and right-hand limits?
8. What are other examples of piecewise functions with similar discontinuities?

### Tip:
Always check both the left-hand and right-hand limits when dealing with piecewise functions to determine if a limit exists at a given point.

Explore the discontinuity of the piecewise function f(x) = x^3 at x = 1. This detailed analysis covers left-hand and right-hand limits, the concept of discontinuity, and implications of different function values and limits. Suitable for high school students and anyone interested in calculus concepts.

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