If f(x)`{:{(x^(2)+4," for "x lt 2 ),(x^(3)," for " x gt 2 ):}` , find `Lim_(x to 2) f(x)`

Answer

To find the limit of the function \( f(x) \) as \( x \) approaches 2, we need to evaluate the left-hand limit and the right-hand limit separately.

Given:
\[ f(x) = \begin{cases} x^2 + 4 & \text{for } x < 2 \\ x^3 & \text{for } x > 2 \end{cases} \]

### Step 1: Find the Left-Hand Limit
We need to find:
\[ \lim_{x \to 2^-} f(x) \]
Since \( x \) is approaching 2 from the left, we use the expression for \( f(x) \) when \( x < 2 \):
\[ f(x) = x^2 + 4 \]
Thus,
\[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2} (x^2 + 4) \]
Substituting \( x = 2 \):
\[ = 2^2 + 4 = 4 + 4 = 8 \]

### Step 2: Find the Right-Hand Limit
Next, we find:
\[ \lim_{x \to 2^+} f(x) \]
Since \( x \) is approaching 2 from the right, we use the expression for \( f(x) \) when \( x > 2 \):
\[ f(x) = x^3 \]
Thus,
\[ \lim_{x \to 2^+} f(x) = \lim_{x \to 2} (x^3) \]
Substituting \( x = 2 \):
\[ = 2^3 = 8 \]

### Step 3: Compare the Limits
Now we compare the left-hand limit and the right-hand limit:
\[ \lim_{x \to 2^-} f(x) = 8 \quad \text{and} \quad \lim_{x \to 2^+} f(x) = 8 \]

Since both limits are equal, we can conclude:
\[ \lim_{x \to 2} f(x) = 8 \]

### Final Answer
Thus, the limit of \( f(x) \) as \( x \) approaches 2 is:
\[ \boxed{8} \]