Math Problem Statement
Where is it continuous?
Solution
To determine where the function is continuous, let's analyze the piecewise definition provided in the image:
\frac{x-2}{x^3 - 8} & \text{if } x \neq 2 \\ 12 & \text{if } x = 2 \end{cases}$$ ### Steps to determine continuity: 1. **Continuity for $$x \neq 2$$:** The function $$k(x)$$ is continuous for all $$x$$ except possibly at $$x = 2$$. The first piece $$\frac{x-2}{x^3 - 8}$$ is continuous wherever it is defined. Notice that $$x^3 - 8 = (x - 2)(x^2 + 2x + 4)$$. Hence, the function can be rewritten as: $$\frac{x-2}{(x-2)(x^2 + 2x + 4)} = \frac{1}{x^2 + 2x + 4}$$ for $$x \neq 2$$. Since $$x^2 + 2x + 4$$ is always positive and never zero for all real $$x$$, the function is continuous for all $$x \neq 2$$. 2. **Continuity at $$x = 2$$:** For the function to be continuous at $$x = 2$$, the following must hold: $$\lim_{x \to 2} k(x) = k(2)$$ We already know $$k(2) = 12$$. Now, let's compute the limit as $$x$$ approaches 2: $$\lim_{x \to 2} \frac{1}{x^2 + 2x + 4} = \frac{1}{2^2 + 2(2) + 4} = \frac{1}{4 + 4 + 4} = \frac{1}{12}$$ Since the limit as $$x$$ approaches 2 is $$\frac{1}{12}$$ and $$k(2) = 12$$, these are not equal. Therefore, $$k(x)$$ is **not** continuous at $$x = 2$$. ### Conclusion: The function $$k(x)$$ is continuous for all $$x$$ except at $$x = 2$$. Would you like to go over the concepts in more detail or ask any questions? Here are some related questions for further exploration: 1. What is the definition of continuity at a point? 2. How do you find the limit of a piecewise function at a point where the pieces meet? 3. Why does the factorization of $$x^3 - 8$$ simplify the function $$k(x)$$? 4. What happens to the continuity if $$k(x)$$ were defined differently at $$x = 2$$? 5. Can you give an example of a function that is continuous everywhere? 6. What is the difference between continuity and differentiability? 7. How can a function be made continuous by redefining a specific point? 8. What role does the intermediate value theorem play in analyzing the continuity of functions? **Tip:** When checking for continuity, always examine the function at the points where the definition changes or where the function might not be well-defined.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
Limit definition
Piecewise function definition
Theorems
Definition of continuity
Suitable Grade Level
Grades 11-12