Math Problem Statement
solve this and also make its graph
Solution
Here is the graph of the function defined by:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Limits
Discontinuity
Formulas
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Theorems
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Suitable Grade Level
High School