https://file.aiquickdraw.com/mathbot/file/1729881808034031sigzv.jpg

The function \( f(x) \) defined in the image appears to be piecewise as follows:

\[
f(x) = 
\begin{cases} 
\frac{1}{e^{\arctan(x)}}, & x < 0 \\
\frac{\sqrt{1 - \sin\left(\frac{\pi x}{2}\right)}}{\pi - x}, & x > 0 
\end{cases}
\]

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 \( x_0 = 0 \)? 

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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.

Find the limits of the piecewise function given by f(x) for x < 0 and x > 0.

Explore the limits and continuity of a piecewise function defined for x < 0 and x > 0. Learn how to calculate the left-hand and right-hand limits to determine the behavior at the point of discontinuity.

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