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The problem asks to determine the left-hand limit of a piecewise function as \( x \to 0^- \).

The given function is:

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

### Solution:

To find the limit when \( x \to 0^- \), we consider the function's value when \( x \) is approaching 0 from the left. For \( x < 0 \), the function \( f(x) = -1 \).

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

Thus, the left-hand limit of the function as \( x \to 0^- \) is \(-1\).

Would you like to ask more questions or get additional details? Here are some related questions to consider:

1. What is the right-hand limit of the function as \( x \to 0^+ \)?
2. Is the function continuous at \( x = 0 \)?
3. What is the overall limit as \( x \to 0 \)?
4. How does the function behave as \( x \to \infty \)?
5. What is the derivative of the function in its different regions?

**Tip:** When dealing with piecewise functions, always analyze the behavior in each region separately to determine limits and continuity.

Considere la función: \( f(x) = \begin{cases} -1 & x < 0 \\ 0 & x = 0 \\ x + 1 & x > 0 \end{cases} \). Determine el límite cuando \( x \to 0^- \).

Determine the left-hand limit of the piecewise function as x approaches 0 from the left. Explore key mathematical concepts, such as limits, continuity, and piecewise function behavior. Suitable for Grades 10-12 students studying calculus.

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