https://file.aiquickdraw.com/mathbot/file/1727482122026guxt18pg.jpg

Let's break down the problems from the image you've uploaded.

### a) The function \( f \) has the following properties:
- \( f(0) = 1 \)
- \( f(1) = -1 \)
- \( f(-1) = 2 \)
- \( f(2) = 0 \)

We are asked to evaluate the composition \( f(f(f(0))) \).

#### Solution:
1. Start by finding \( f(0) \), which is given:
   \[
   f(0) = 1
   \]
2. Now find \( f(f(0)) = f(1) \). From the problem, we know:
   \[
   f(1) = -1
   \]
3. Finally, find \( f(f(f(0))) = f(-1) \). We are given:
   \[
   f(-1) = 2
   \]
Thus, the value of \( f(f(f(0))) \) is \( 2 \).

### b) The function \( f(x) \) is defined as:
\[
f(x) = \begin{cases} 
x, & \text{if } x \geq 0 \\
1, & \text{if } x < 0 
\end{cases}
\]
We are asked to find the limit:
\[
\lim_{x \to 0} f(x)
\]

#### Solution:
To evaluate this limit, we must examine the left-hand limit (as \( x \to 0^- \)) and the right-hand limit (as \( x \to 0^+ \)) separately.

1. **Right-hand limit** \( \lim_{x \to 0^+} f(x) \):
   - For \( x \to 0^+ \) (values of \( x \) approaching 0 from the positive side), \( f(x) = x \). So:
     \[
     \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0
     \]

2. **Left-hand limit** \( \lim_{x \to 0^-} f(x) \):
   - For \( x \to 0^- \) (values of \( x \) approaching 0 from the negative side), \( f(x) = 1 \). So:
     \[
     \lim_{x \to 0^-} f(x) = 1
     \]

Since the left-hand and right-hand limits are not equal, the limit \( \lim_{x \to 0} f(x) \) does **not exist**.

### Summary:
- For part (a), \( f(f(f(0))) = 2 \).
- For part (b), the limit \( \lim_{x \to 0} f(x) \) does not exist because the left-hand and right-hand limits are different.

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Do you need further details or explanations on any part? Here are some related questions to expand your understanding:

1. How do you evaluate composite functions step by step?
2. What conditions are necessary for a limit to exist at a point?
3. What does it mean for a function to be continuous at a point?
4. How can you graphically interpret the difference between left-hand and right-hand limits?
5. Can a function be discontinuous at a point if both the left-hand and right-hand limits exist but are different?

**Tip:** When finding limits, always check both the left-hand and right-hand limits separately for functions that have different behaviors on either side of a point.

A function f has the property that f(0) = 1, f(1) = -1, f(-1) = 2, and f(2) = 0. Evaluate the composition f(f(f(0))). Let f(x) be defined by f(x) = x, if x >= 0, and f(x) = 1, if x < 0. Find the limit of f(x) as x approaches 0.

Solve a problem involving function composition and evaluate the limit of a piecewise function as x approaches 0. Explore key concepts in calculus, including function properties and the limit theorem. Suitable for students in Grades 11-12.

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