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The given piecewise function \( f(x) \) is defined as:

\[
f(x) = 
\begin{cases} 
\sqrt{x - 4} & \text{if } x > 4 \\
2x - 8 & \text{if } 0 \leq x \leq 4 \\
-1 & \text{if } x < 0 
\end{cases}
\]

To find the limit of \( f(x) \) as \( x \) approaches 0, we need to consider the left-hand limit (as \( x \) approaches 0 from the left) and the right-hand limit (as \( x \) approaches 0 from the right).

- **For \( x < 0 \)**: \( f(x) = -1 \). Therefore, the left-hand limit as \( x \) approaches 0 is:
  \[
  \lim_{x \to 0^-} f(x) = -1
  \]

- **For \( 0 \leq x \leq 4 \)**: \( f(x) = 2x - 8 \). Therefore, the right-hand limit as \( x \) approaches 0 is:
  \[
  \lim_{x \to 0^+} f(x) = 2(0) - 8 = -8
  \]

Since the left-hand limit and right-hand limit are not equal (\(-1 \neq -8\)), the limit of \( f(x) \) as \( x \) approaches 0 does not exist.

Would you like further details or have any questions? Here are some related questions you might consider:

1. What are the limits of \( f(x) \) as \( x \) approaches 4 from both sides?
2. How can we find the continuity of \( f(x) \) at \( x = 4 \)?
3. What does it mean for a function to have a limit at a point?
4. Can a function be continuous at a point where its limit does not exist?
5. How does piecewise function behavior affect the existence of limits?

**Tip:** When dealing with piecewise functions, always check the behavior of the function from both sides of the point of interest to determine the existence of the limit.

Explore the concept of limits in piecewise functions with a focus on the example where the function does not have a limit at x = 0. Learn about left-hand and right-hand limits, their calculation, and the implications of piecewise function behavior on limit existence.

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