https://file.aiquickdraw.com/mathbot/file/1725926653491taaisdo3.jpg

Let's break down the problem step by step.

### Given Function:
\[
f(x) =
\begin{cases}
4 - x^2, & \text{if } x \neq -2 \\
7, & \text{if } x = -2
\end{cases}
\]

---

### Part (a) - Graph \( f(x) \)

The function is piecewise:

- For \( x \neq -2 \), the function is \( f(x) = 4 - x^2 \), a parabola opening downward.
- For \( x = -2 \), the function is \( f(x) = 7 \), which creates a single point at \( (-2, 7) \).

When graphing:

- Plot the parabola \( f(x) = 4 - x^2 \), which is symmetric around the y-axis.
- At \( x = -2 \), instead of following the parabola's value at \( x = -2 \), place a dot at \( (-2, 7) \).

---

### Part (b) - Limits as \( x \to -2^- \) and \( x \to -2^+ \)

#### Left-hand limit (\( x \to -2^- \)):
This is the limit of \( f(x) = 4 - x^2 \) as \( x \) approaches \( -2 \) from the left.

\[
\lim_{x \to -2^-} f(x) = 4 - (-2)^2 = 4 - 4 = 0
\]

#### Right-hand limit (\( x \to -2^+ \)):
This is the limit of \( f(x) = 4 - x^2 \) as \( x \) approaches \( -2 \) from the right.

\[
\lim_{x \to -2^+} f(x) = 4 - (-2)^2 = 0
\]

Thus, both the left-hand and right-hand limits are equal to 0.

---

### Part (c) - Does \( \lim_{x \to -2} f(x) \) exist?

The overall limit exists if both one-sided limits are equal. From part (b), we found:

\[
\lim_{x \to -2^-} f(x) = \lim_{x \to -2^+} f(x) = 0
\]

However, \( f(x) \) is defined as 7 when \( x = -2 \), which causes a discontinuity. Therefore, even though the limits from both sides are the same, the function's value at \( x = -2 \) differs from this limit.

Thus, \( \lim_{x \to -2} f(x) = 0 \), but \( f(-2) = 7 \). Therefore, the limit exists, but the function is not continuous at \( x = -2 \).

---

Would you like a more detailed graph, or do you have any other questions?

### Related Questions:
1. What is the difference between a limit and the actual value of a function at a point?
2. How do you determine continuity at a point for a piecewise function?
3. What happens when one-sided limits are not equal?
4. How do you graph piecewise functions with discontinuities?
5. Can a function have a limit but still be discontinuous at that point?

### Tip:
When graphing a piecewise function, always pay attention to the point where the function changes its definition, as it often creates discontinuities or special points.

a. Graph f(x) = { 4 - x^2, x ≠ -2; 7, x = -2 } b. Find lim f(x) as x approaches -2 from the left and right. c. Does lim f(x) exist as x approaches -2?

Explore a piecewise function with a discontinuity at x = -2. Learn how to find limits from both sides and understand the concept of discontinuity. This problem helps students master piecewise functions, limit evaluation, and discontinuity.

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