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The problem asks if the function \( f(x) \) is continuous at \( x = -1 \) and to use the 3-step definition of continuity to justify the answer.

The function \( f(x) \) is piecewise, defined as follows:

\[
f(x) = \begin{cases} 
x + 3, & \text{if } x < -1 \\
2, & \text{if } x = -1 \\
\frac{2}{1 + e^{1 + x}}, & \text{if } x > -1 
\end{cases}
\]

### Steps for Checking Continuity at \( x = -1 \)

To determine whether \( f(x) \) is continuous at \( x = -1 \), we need to use the 3-step definition of continuity:

1. **Check if \( f(-1) \) is defined**:  
   We need to verify that the function has a value at \( x = -1 \).  
   From the piecewise definition, \( f(-1) = 2 \).

2. **Check if the limit of \( f(x) \) as \( x \) approaches \( -1 \) from both sides exists**:
   - **Left-hand limit**:  
     \( \lim_{x \to -1^-} f(x) \) means \( x \) approaches \( -1 \) from the left (i.e., \( x < -1 \)). The expression for this is \( f(x) = x + 3 \). So:
     \[
     \lim_{x \to -1^-} f(x) = (-1) + 3 = 2
     \]
   - **Right-hand limit**:  
     \( \lim_{x \to -1^+} f(x) \) means \( x \) approaches \( -1 \) from the right (i.e., \( x > -1 \)). The expression for this is \( f(x) = \frac{2}{1 + e^{1 + x}} \). Let's compute this limit as \( x \to -1^+ \):
     \[
     \lim_{x \to -1^+} \frac{2}{1 + e^{1 + x}} = \frac{2}{1 + e^0} = \frac{2}{1 + 1} = 1
     \]

3. **Check if the limit equals the function value at \( x = -1 \)**:  
   For the function to be continuous at \( x = -1 \), we need the left-hand limit, right-hand limit, and \( f(-1) \) to all be equal.  
   - Left-hand limit: 2  
   - Right-hand limit: 1  
   - \( f(-1) = 2 \)

Since the right-hand limit does not equal the left-hand limit (2 ≠ 1), the function is **not continuous** at \( x = -1 \).

Would you like more details on any step or further clarification?

### Related Questions:
1. What is the significance of left-hand and right-hand limits in continuity?
2. How do you compute the limit of an exponential function as \( x \to -1 \)?
3. Can a function be continuous if only one of the one-sided limits equals the function value?
4. What types of functions are typically discontinuous at a point?
5. How does the definition of continuity extend to higher dimensions?

**Tip**: When checking continuity at a point for piecewise functions, always carefully evaluate the limit from both sides.

Is f(x) continuous at x = -1? Show the 3-step definition of continuity to justify your answer.

This problem involves verifying the continuity of a piecewise function at x = -1 using the 3-step definition of continuity. The solution covers limit calculations from both sides and evaluates the function value at the point.

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