Math Problem Statement
4 pts) For what value(s) of the constants ππ is the function ππ continuous at π₯π₯ = 0? ππ(π₯π₯) = οΏ½ ππ(π₯π₯2 + 2ππ) ππππ π₯π₯ < 0 4 + ππβπ₯π₯ + 4 ππππ π₯π₯ β₯ 0
Solution
To determine for what values of the function is continuous at , we need to ensure the left-hand limit, the right-hand limit, and the value of the function at are all equal.
The given function is:
\begin{cases} a(x^2 + 2a) & \text{if } x < 0 \\ 4 + a\sqrt{x} + 4 & \text{if } x \geq 0 \end{cases}$$ ### Step 1: Evaluate $$f(0)$$ From the definition of $$f(x)$$ when $$x \geq 0$$, the value of the function at $$x = 0$$ is: $$f(0) = 4 + a\sqrt{0} + 4 = 4 + 0 + 4 = 8$$ So, $$f(0) = 8$$. ### Step 2: Left-hand limit $$\lim_{x \to 0^-} f(x)$$ For $$x < 0$$, we use the expression $$f(x) = a(x^2 + 2a)$$. As $$x \to 0^-$$, this becomes: $$\lim_{x \to 0^-} f(x) = a(0^2 + 2a) = a(2a) = 2a^2$$ ### Step 3: Right-hand limit $$\lim_{x \to 0^+} f(x)$$ For $$x \geq 0$$, we use the expression $$f(x) = 4 + a\sqrt{x} + 4$$. As $$x \to 0^+$$, this becomes: $$\lim_{x \to 0^+} f(x) = 4 + a\sqrt{0} + 4 = 8$$ ### Step 4: Continuity condition For the function to be continuous at $$x = 0$$, the left-hand limit, right-hand limit, and $$f(0)$$ must all be equal. Thus, we require: $$2a^2 = 8$$ Solving for $$a$$: $$a^2 = 4 \quad \Rightarrow \quad a = \pm 2$$ ### Conclusion: The function $$f(x)$$ is continuous at $$x = 0$$ if $$a = 2$$ or $$a = -2$$. Would you like further details or have any questions? Here are five related questions: 1. How do you verify if a function is continuous at a point? 2. What happens if the limits from the left and right do not match at a point of interest? 3. How do square roots affect continuity conditions? 4. Can you describe why piecewise functions require checking limits at the transition point? 5. How would the problem change if the expression for $$f(x)$$ were different for $$x \geq 0$$? **Tip**: When solving continuity problems, always check both the left-hand and right-hand limits along with the value of the function at the point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
Limit formula: lim_{x -> 0^-} f(x), lim_{x -> 0^+} f(x)
Continuity condition: f(0) = lim_{x -> 0^-} f(x) = lim_{x -> 0^+} f(x)
Theorems
Continuity Theorem: A function is continuous at a point if the limit from the left, limit from the right, and the value of the function at that point are equal.
Suitable Grade Level
Grades 11-12
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