https://file.aiquickdraw.com/mathbot/file/1725991360886snn4z31q.jpg

Let's break down and solve each part of the problem from the image.

### The function \( f(x) \) is defined as:
\[
f(x) = 
\begin{cases}
-2x - 4 & \text{if } x \leq -4, \\
4 & \text{if } -4 < x < -2, \\
4 - x^2 & \text{if } x \geq -2.
\end{cases}
\]

### a) \( f(-2) \):
To find \( f(-2) \), we look at the third case \( f(x) = 4 - x^2 \), which applies when \( x \geq -2 \). Thus:
\[
f(-2) = 4 - (-2)^2 = 4 - 4 = 0.
\]
So, \( f(-2) = 0 \).

### b) \( \lim_{x \to -2^-} f(x) \):
We need to find the left-hand limit as \( x \to -2 \) from the left. For \( x < -2 \), the second piece of the function applies, where \( f(x) = 4 \). Therefore:
\[
\lim_{x \to -2^-} f(x) = 4.
\]

### c) \( \lim_{x \to -2^+} f(x) \):
We need to find the right-hand limit as \( x \to -2 \) from the right. For \( x \geq -2 \), the third piece applies, where \( f(x) = 4 - x^2 \). As \( x \to -2^+ \):
\[
\lim_{x \to -2^+} f(x) = 4 - (-2)^2 = 4 - 4 = 0.
\]

### d) \( \lim_{x \to -2} f(x) \):
For the two-sided limit to exist, the left-hand limit and right-hand limit must be equal. From parts (b) and (c):
\[
\lim_{x \to -2^-} f(x) = 4 \quad \text{and} \quad \lim_{x \to -2^+} f(x) = 0.
\]
Since these limits are not equal, the two-sided limit does not exist:
\[
\lim_{x \to -2} f(x) \quad \text{does not exist}.
\]

### e) \( f(-4) \):
To find \( f(-4) \), we use the first case since \( x = -4 \), which is included in the \( x \leq -4 \) range. Thus:
\[
f(-4) = -2(-4) - 4 = 8 - 4 = 4.
\]

### f) \( \lim_{x \to -4^-} f(x) \):
For \( x \to -4^- \), we are approaching from the left, so we use the first case \( f(x) = -2x - 4 \). Therefore:
\[
\lim_{x \to -4^-} f(x) = -2(-4) - 4 = 8 - 4 = 4.
\]

### g) \( \lim_{x \to -4^+} f(x) \):
For \( x \to -4^+ \), we are approaching from the right. In this interval \( -4 < x < -2 \), the function is constant at \( f(x) = 4 \). Thus:
\[
\lim_{x \to -4^+} f(x) = 4.
\]

### h) \( \lim_{x \to -4} f(x) \):
Since both the left-hand limit and right-hand limit at \( x = -4 \) are equal:
\[
\lim_{x \to -4} f(x) = 4.
\]

### Summary:
- a) \( f(-2) = 0 \)
- b) \( \lim_{x \to -2^-} f(x) = 4 \)
- c) \( \lim_{x \to -2^+} f(x) = 0 \)
- d) \( \lim_{x \to -2} f(x) \) does not exist.
- e) \( f(-4) = 4 \)
- f) \( \lim_{x \to -4^-} f(x) = 4 \)
- g) \( \lim_{x \to -4^+} f(x) = 4 \)
- h) \( \lim_{x \to -4} f(x) = 4 \)

Would you like a graph of the function or any further clarifications? Here are 5 related questions:

1. How do left-hand and right-hand limits differ?
2. What happens if the two-sided limit exists but is different from the function value?
3. How does continuity relate to these limit evaluations?
4. Can you explain the concept of piecewise functions in more detail?
5. What is the relationship between differentiability and continuity at a point?

**Tip**: When dealing with limits, always check left-hand and right-hand limits to determine if a two-sided limit exists!

Sketch the graph of f(x) and evaluate the given limits and function values based on the piecewise function provided.

Learn how to evaluate the limits and specific values of a piecewise function, covering concepts such as left-hand and right-hand limits, continuity, and function evaluation. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation