Math Problem Statement
Solution
The problem asks us to determine the value of such that the given function is continuous at . The function is piecewise, and it is defined as:
\begin{cases} \frac{2x + \sin^{-1}(x)}{2x - \tan^{-1}(x)}, & x \neq 0 \\ 2\phi - 1, & x = 0 \end{cases}$$ To ensure the function is continuous at $$x = 0$$, we need to make sure that: $$\lim_{x \to 0} f(x) = f(0)$$ ### Step 1: Find $$\lim_{x \to 0} f(x)$$ for $$x \neq 0$$ We will compute the limit of the expression for $$f(x)$$ as $$x \to 0$$: $$\lim_{x \to 0} \frac{2x + \sin^{-1}(x)}{2x - \tan^{-1}(x)}$$ For small values of $$x$$, the approximations $$\sin^{-1}(x) \approx x$$ and $$\tan^{-1}(x) \approx x$$ hold. Applying these approximations: $$\lim_{x \to 0} \frac{2x + x}{2x - x} = \lim_{x \to 0} \frac{3x}{x} = 3$$ Thus: $$\lim_{x \to 0} f(x) = 3$$ ### Step 2: Continuity condition at $$x = 0$$ For $$f(x)$$ to be continuous at $$x = 0$$, we must have: $$f(0) = \lim_{x \to 0} f(x)$$ Given that $$f(0) = 2\phi - 1$$, we set: $$2\phi - 1 = 3$$ ### Step 3: Solve for $$\phi$$ Now, solve for $$\phi$$: $$2\phi = 4 \quad \Rightarrow \quad \phi = 2$$ Thus, the value of $$\phi$$ that ensures continuity at $$x = 0$$ is $$\boxed{2}$$. --- Do you want further clarification or details on any part of this solution? Here are some related questions to explore: 1. What are some other common methods for ensuring the continuity of piecewise functions? 2. Why do we use small approximations like $$\sin^{-1}(x) \approx x$$ as $$x \to 0$$? 3. What would happen if we chose a different value of $$\phi$$? 4. How do limits help in defining the continuity of a function? 5. Can a function be continuous but not differentiable? Can you provide an example? **Tip:** Always check for both one-sided limits when determining the continuity of a piecewise function at a specific point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity of Piecewise Functions
Trigonometric Approximations
Formulas
lim_{x \to 0} \frac{2x + \sin^{-1}(x)}{2x - \tan^{-1}(x)}
Continuity condition: f(0) = \lim_{x \to 0} f(x)
Theorems
Limit Theorem
Continuity Theorem for Piecewise Functions
Suitable Grade Level
College Level Calculus
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