Math Problem Statement
Find the limits of the piecewise function given by f(x) for x < 0 and x > 0.
Solution
The function defined in the image appears to be piecewise as follows:
\begin{cases} \frac{1}{e^{\arctan(x)}}, & x < 0 \\ \frac{\sqrt{1 - \sin\left(\frac{\pi x}{2}\right)}}{\pi - x}, & x > 0 \end{cases}$$ There are two parts of the problem related to calculating limits: 1. $$\lim_{x \to x_0^+} f(x)$$ 2. $$\lim_{x \to x_0^-} f(x)$$ Where $$x_0$$ is likely the point of interest where the continuity or type of discontinuity will be examined. Let me know if you'd like me to solve for specific points, such as $$x_0 = 0$$, or if you'd like to explore how to handle the limits from both sides to check the function's continuity or discontinuity. Would you like to proceed with analyzing the function at $$x_0 = 0$$? --- Here are five follow-up questions for this type of analysis: 1. What happens to the function at $$x_0 = 0$$? Is it continuous there? 2. How do we determine if a discontinuity is removable? 3. What is the behavior of the left-hand limit $$\lim_{x \to 0^-} f(x)$$? 4. What is the behavior of the right-hand limit $$\lim_{x \to 0^+} f(x)$$? 5. How does the arctangent function behave as $$x \to 0^-$$? **Tip:** When dealing with piecewise functions, always check the limit from both sides of the point of interest to determine if the function is continuous or has a discontinuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise functions
Continuity
Discontinuities
Formulas
f(x) = { 1/(e^(arctan(x))), x < 0 }
f(x) = sqrt(1 - sin(pi * x / 2)) / (pi - x), x > 0
Theorems
Limit definition
Continuity definition
Removable discontinuity
Suitable Grade Level
University