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To find the limit \(\lim_{x \to 0} f(x)\) where \( f(x) \) is defined as:

\[
f(x) = 
\begin{cases}
x + 3, & \text{if } x < 0 \\
-x + 3, & \text{if } x \geq 0
\end{cases}
\]

we need to evaluate the left-hand limit (\(x \to 0^-\)) and the right-hand limit (\(x \to 0^+\)) separately.

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

For \(x < 0\), the function is given by \(f(x) = x + 3\).

\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x + 3) = 0 + 3 = 3
\]

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

For \(x \geq 0\), the function is given by \(f(x) = -x + 3\).

\[
\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (-x + 3) = -0 + 3 = 3
\]

Since both the left-hand limit and the right-hand limit are equal:

\[
\lim_{x \to 0} f(x) = 3
\]

### Conclusion:
\[
\lim_{x \to 0} f(x) = 3
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you determine continuity at a point using limits?
2. What happens if the left-hand and right-hand limits are not equal?
3. How do you graph piecewise functions and determine their limits visually?
4. Can limits exist even if a function is undefined at that point?
5. How do limits relate to derivatives in calculus?

### Tip:
Always check the behavior of a function from both sides of the point when dealing with piecewise functions to ensure accurate limit evaluation.

Learn how to find the limit of a piecewise function as x approaches 0, where the function changes at x = 0. This step-by-step guide covers evaluating both the left-hand and right-hand limits, resulting in a limit of 3. Ideal for college-level calculus students.

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