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The problem asks you to ensure that the function \( f(x) \) is continuous at \( x = -1 \) and also involves evaluating various limits for \( f(x) \). The piecewise function is defined as:

\[
f(x) =
\begin{cases}
4 - x^2, & x < -1 \\
ax^2 + 2, & x \geq -1
\end{cases}
\]

### Part (a): Find \( a \) to make \( f(x) \) continuous at \( x = -1 \).

To ensure continuity at \( x = -1 \), the left-hand limit and the right-hand limit at \( x = -1 \) must be equal, and this value must also be equal to \( f(-1) \). 

1. **Left-hand limit** as \( x \to -1^- \) (using the first part of the function):
   \[
   \lim_{x \to -1^-} f(x) = 4 - (-1)^2 = 4 - 1 = 3
   \]

2. **Right-hand limit** as \( x \to -1^+ \) (using the second part of the function):
   \[
   \lim_{x \to -1^+} f(x) = a(-1)^2 + 2 = a(1) + 2 = a + 2
   \]

For continuity, we require:
\[
3 = a + 2
\]

Solving for \( a \):
\[
a = 1
\]

So, \( a = 1 \) makes the function continuous at \( x = -1 \).

---

### Part (b)-(e): Evaluate each limit.

- **(b) \( \lim_{x \to -1} f(x) \)**:
  Since the function is continuous at \( x = -1 \), the limit as \( x \to -1 \) is simply \( f(-1) \), which we computed as 3. So,
  \[
  \lim_{x \to -1} f(x) = 3
  \]

- **(c) \( \lim_{x \to 0} f(x) \)**:
  For \( x = 0 \), use the second piece \( f(x) = ax^2 + 2 \) with \( a = 1 \):
  \[
  \lim_{x \to 0} f(x) = (1)(0^2) + 2 = 2
  \]

- **(d) \( \lim_{x \to 2} f(x) \)**:
  For \( x = 2 \), use the second part \( f(x) = x^2 + 2 \):
  \[
  \lim_{x \to 2} f(x) = (1)(2^2) + 2 = 4 + 2 = 6
  \]

- **(e) \( \lim_{x \to \infty} f(x) \)**:
  For \( x \to \infty \), the second part of the function dominates:
  \[
  \lim_{x \to \infty} f(x) = \lim_{x \to \infty} (x^2 + 2) = \infty
  \]

Would you like more details on any part of this problem?

---

Here are 5 related questions to expand understanding:

1. What is the formal definition of continuity at a point?
2. Why do we require both the left-hand and right-hand limits to match for continuity?
3. How would the function change if \( a \neq 1 \)?
4. What would happen if the function had a different behavior for \( x > -1 \)?
5. How do you approach limits at infinity for different types of functions?

### Tip: Always check both the left-hand and right-hand limits to determine continuity at a point, especially for piecewise functions.

Suppose that the function f(x) is continuous at x = -1 and that f(x) is defined by f(x) = { 4 - x^2, x < -1 ; ax^2 + 2, x >= -1 }. (a) Find the value of a to make f(x) continuous. (b) Evaluate lim_{x -> -1} f(x). (c) Evaluate lim_{x -> 0} f(x). (d) Evaluate lim_{x -> 2} f(x). (e) Evaluate lim_{x -> infinity} f(x).

This problem explores the continuity of a piecewise function and evaluates several limits, including at -1, 0, 2, and infinity. It involves solving for a parameter to ensure the function is continuous at a specific point. Suitable for calculus students.

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