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To solve this problem, we need to evaluate the limit of \( f(x) \) as \( x \) approaches \( 2 \) from the left, which means we need to use the expression defined for \( x < 2 \). The function \( f(x) \) is defined as:

\[
f(x) = \begin{cases} 
1 + x^2 & \text{if } x < 2 \\
x^3 & \text{if } x \geq 2 
\end{cases}
\]

Since we are looking for the limit from the left side, we will use the expression \( f(x) = 1 + x^2 \) as \( x \to 2^- \).

### Step-by-Step Solution

1. **Substitute \( x = 2 \) in the expression \( 1 + x^2 \):**
   \[
   f(x) = 1 + x^2
   \]

2. **Calculate \( f(2) \) using this expression:**
   \[
   f(2) = 1 + (2)^2 = 1 + 4 = 5
   \]

Thus, the limit of \( f(x) \) as \( x \) approaches \( 2 \) from the left is \( 5 \).

### Answer
The correct answer is **(a) 5**.

Would you like further details on this calculation or have any questions?

### Related Questions
1. What is the limit of \( f(x) \) as \( x \) approaches \( 2 \) from the right?
2. How does the definition of the function change for values of \( x \) greater than or equal to 2?
3. Why is it necessary to approach from the left to use the expression \( 1 + x^2 \)?
4. What is the overall limit of \( f(x) \) as \( x \) approaches \( 2 \) (if it exists)?
5. How would the limit change if the function were defined differently for \( x < 2 \)?

### Tip
When finding limits from one side, always ensure you're using the correct part of the function's piecewise definition.

Find the limit of \( f(x) \) as \( x \) tends to 2 from the left if \( f(x) = \begin{cases} 1 + x^2 & \text{if } x < 2 \\ x^3 & \text{if } x \geq 2 \end{cases} \).

Understand how to find the limit of a piecewise function as x approaches 2 from the left. This question involves evaluating limits for piecewise-defined functions, ideal for high school calculus students in Grades 11-12.

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