https://file.aiquickdraw.com/mathbot/file/1725536646815cj1res98.jpg

To analyze the given limit, let's break it down step by step.

We are asked to evaluate:

\[
\lim_{x \to 4^-} f(x)
\]

Where the function \( f(x) \) is piecewise defined as:

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

Now, let's analyze what happens as \( x \to 4^- \) (from the left side):

- For \( 1 \leq x < 4 \), the function \( f(x) = 1 \). This means that as \( x \) approaches 4 from values less than 4 but greater than or equal to 1, \( f(x) \) is constantly equal to 1.

Therefore:

\[
\lim_{x \to 4^-} f(x) = 1
\]

The correct answer is:

\[
\boxed{1}
\]

Would you like further details on how the piecewise function behaves graphically or any other aspects of the solution?

Here are five related questions to explore:

1. What is the limit of \( f(x) \) as \( x \to 4^+ \)?
2. How does the function behave as \( x \to 0 \)?
3. What would the graph of \( \sqrt{1 - x^2} \) look like for \( 0 \leq x < 1 \)?
4. What is the importance of continuity when dealing with piecewise functions?
5. How would the limit change if the piecewise function was different at \( x = 4 \)?

**Tip**: When analyzing limits, always pay attention to whether you're approaching from the left (denoted \( x \to c^- \)) or from the right (denoted \( x \to c^+ \)), especially with piecewise functions.

Determine the limit by sketching an appropriate graph: 

\(
\lim_{x \to 4^-} f(x)
\)

where \( f(x) \) is defined as:

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

Options: 
- 0
- 1
- 4
- Does not exist

Explore how to determine the limit of a piecewise function as x approaches 4 from the left. This detailed guide includes an analysis of limits, piecewise functions, and continuity principles, suitable for high school students in Grades 10-12.

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