Math Problem Statement
For what values of a, if any, is the function f(x) continuous at x = 3?
Solution
To determine the value of such that the function is continuous at , we need to ensure that the left-hand limit, the right-hand limit, and the function's value at all match.
Given function:
\begin{cases} a^2 + x^2 & \text{for } x < 3, \\ a(x + 3) & \text{for } x \geq 3. \end{cases}$$ ### Continuity condition at $$x = 3$$: For continuity at $$x = 3$$, the left-hand limit as $$x \to 3^-$$, the right-hand limit as $$x \to 3^+$$, and $$f(3)$$ must all be equal. 1. **Left-hand limit $$\lim_{x \to 3^-} f(x)$$**: From the definition of $$f(x)$$ when $$x < 3$$: $$\lim_{x \to 3^-} f(x) = a^2 + 3^2 = a^2 + 9.$$ 2. **Right-hand limit $$\lim_{x \to 3^+} f(x)$$**: From the definition of $$f(x)$$ when $$x \geq 3$$: $$\lim_{x \to 3^+} f(x) = f(3) = a(3 + 3) = 6a.$$ For $$f(x)$$ to be continuous at $$x = 3$$, we must have: $$a^2 + 9 = 6a.$$ ### Solve the equation: $$a^2 - 6a + 9 = 0.$$ This is a quadratic equation: $$(a - 3)^2 = 0.$$ Thus, $$a = 3$$. ### Conclusion: The function is continuous at $$x = 3$$ if and only if $$a = 3$$. So, the correct answer is $$\boxed{B}$$. --- Do you need further details or have any questions? Here are 5 related questions to explore: 1. What is the definition of continuity at a point? 2. How do you find the limit of a piecewise function? 3. What happens if $$a$$ takes other values in this function? 4. Can a function be discontinuous at multiple points and still be differentiable? 5. How would the function's continuity change if the piecewise conditions were defined for different ranges? **Tip:** Continuity at a point requires the function value and both one-sided limits to be equal. Always check for this condition when evaluating continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Quadratic equation formula: ax^2 + bx + c = 0
Limit formulas for continuity at a point
Theorems
Continuity Theorem
Quadratic Formula
Suitable Grade Level
Grades 10-12