Math Problem Statement
Solution
We are given a function that is piecewise defined on the interval , and we are asked to determine the constants and such that is continuous at .
The function is given as:
\begin{cases} \frac{3x^2}{\sqrt{x^2 + a^2} - a}, & \text{if } -\frac{\pi}{2} < x < 0 \\ b, & \text{if } x = 0 \\ \frac{b}{3^{1 - \tan x} + 3^{1 - \cot x}}, & \text{if } 0 < x < \frac{\pi}{2} \end{cases}$$ To ensure the continuity of $$f(x)$$ at $$x = 0$$, the left-hand limit $$\lim_{x \to 0^-} f(x)$$, the right-hand limit $$\lim_{x \to 0^+} f(x)$$, and the value of the function at $$x = 0$$, which is $$f(0) = b$$, must all be equal. ### Step 1: Left-hand limit as $$x \to 0^-$$ For $$x < 0$$, we have: $$f(x) = \frac{3x^2}{\sqrt{x^2 + a^2} - a}$$ Taking the limit as $$x \to 0^-$$: $$\lim_{x \to 0^-} f(x) = \frac{3(0)^2}{\sqrt{0^2 + a^2} - a} = 0$$ So the left-hand limit is $$0$$. ### Step 2: Right-hand limit as $$x \to 0^+$$ For $$x > 0$$, we have: $$f(x) = \frac{b}{3^{1 - \tan x} + 3^{1 - \cot x}}$$ Taking the limit as $$x \to 0^+$$, both $$\tan(0) = 0$$ and $$\cot(0) = \infty$$, so: $$\lim_{x \to 0^+} f(x) = \frac{b}{3^{1 - 0} + 3^{1 - \infty}} = \frac{b}{3 + 0} = \frac{b}{3}$$ So the right-hand limit is $$\frac{b}{3}$$. ### Step 3: Continuity at $$x = 0$$ For continuity at $$x = 0$$, the left-hand limit, right-hand limit, and $$f(0) = b$$ must all be equal. Therefore, we set: $$\lim_{x \to 0^-} f(x) = f(0) = \lim_{x \to 0^+} f(x)$$ This gives us the following conditions: 1. $$\lim_{x \to 0^-} f(x) = 0$$ implies $$f(0) = b = 0$$. 2. $$\lim_{x \to 0^+} f(x) = \frac{b}{3}$$, and since $$b = 0$$, this condition is also satisfied. ### Final Answer: The constants are: $$a = 0, \quad b = 0$$ Would you like further clarification or details? Here are 5 questions that build on this: 1. What is the definition of continuity for a piecewise function? 2. How does one compute limits of piecewise functions at a point of interest? 3. Why does setting $$a = 0$$ ensure the continuity for the left-hand limit? 4. What role does the value of $$b$$ play in the right-hand limit? 5. How would the analysis change if the interval for the function were different? **Tip**: For limits involving piecewise functions, always evaluate each piece separately and check continuity by comparing the limits at the boundary points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Left-hand limit: \( \lim_{x \to 0^-} f(x) \)
Right-hand limit: \( \lim_{x \to 0^+} f(x) \)
Continuity condition: \( \lim_{x \to 0^-} f(x) = f(0) = \lim_{x \to 0^+} f(x) \)
Theorems
Continuity of piecewise functions at a point
Limit properties of trigonometric functions
Suitable Grade Level
Undergraduate Calculus or Advanced High School Calculus
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