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The given piecewise function is defined as follows:

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

You are asked to determine the limit as \( x \) approaches \( 0^- \), which means the limit as \( x \) approaches 0 from the left (from negative values of \( x \)).

For \( x < 0 \), the function is defined as \( f(x) = -1 \). Therefore, as \( x \) approaches 0 from the left, the function remains \( -1 \).

Thus, the limit is:

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

Would you like further details, or do you have any questions about this?

Here are five related questions to expand your understanding:
1. What is the limit of \( f(x) \) as \( x \to 0^+ \) (from the right)?
2. Does the limit \( \lim_{x \to 0} f(x) \) exist? Why or why not?
3. How does the function behave for large positive and negative values of \( x \)?
4. Can you sketch the graph of the given piecewise function?
5. How would you interpret the limit behavior at discontinuities?

**Tip**: When dealing with piecewise functions, always check the behavior from both sides of a point to determine limits properly!

Determine el límite cuando x → 0⁻ para la función definida por partes:
 f(x) = 
{-1 si x < 0, 0 si x = 0, x + 1 si x > 0}

Learn how to calculate the limit of a piecewise function as x approaches 0 from the left (x → 0⁻). This example uses a piecewise function defined by different expressions for x < 0, x = 0, and x > 0. Suitable for high school students in Grades 11-12.

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