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The image contains a mathematical problem asking to find the limit of a piecewise function \( f(x) \) as \( x \to 0 \). The function is defined as follows:

\[
f(x) = \begin{cases} 
-1, & x < 0 \\
1, & x > 0
\end{cases}
\]

The problem asks to evaluate the limit:

\[
\lim_{x \to 0} f(x)
\]

### Solution:

To evaluate the limit, we must check the left-hand limit (as \( x \) approaches 0 from the left) and the right-hand limit (as \( x \) approaches 0 from the right).

1. **Left-hand limit (\( x \to 0^- \)):**

   When \( x \to 0^- \), meaning \( x \) approaches 0 from the left (negative side), the function takes the value \( f(x) = -1 \).

   \[
   \lim_{x \to 0^-} f(x) = -1
   \]

2. **Right-hand limit (\( x \to 0^+ \)):**

   When \( x \to 0^+ \), meaning \( x \) approaches 0 from the right (positive side), the function takes the value \( f(x) = 1 \).

   \[
   \lim_{x \to 0^+} f(x) = 1
   \]

### Conclusion:

Since the left-hand limit \( -1 \) and the right-hand limit \( 1 \) are not equal, the two-sided limit does not exist.

Therefore, the limit does not exist:
\[
\lim_{x \to 0} f(x) \text{ does not exist}.
\]

Would you like more details or explanations?

Here are some related questions you might find helpful:
1. What are the conditions for a limit to exist?
2. How do piecewise functions affect limits?
3. Can limits exist if the left and right limits are not equal?
4. What is the difference between one-sided and two-sided limits?
5. How are discontinuities related to limits?

**Tip:** When analyzing limits of piecewise functions, always check both the left-hand and right-hand limits to determine if the overall limit exists.

Find the limit if it exists: \(\lim_{{x \to 0}} f(x)\), where \(f(x) = \begin{cases} -1, & x < 0 \\ 1, & x > 0 \end{cases}\).

Learn how to evaluate the limit of a piecewise function as x approaches 0. This problem involves understanding left-hand and right-hand limits, and the conditions under which a limit does not exist. Suitable for students in Grades 11-12 or early college.

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