`lim_(x to 2) (x^(3) + x^(2) + 4x + 12)/(x^(3) - 3x + 2)` is equal to

Options

A. 8
B. `(3)/(4)`
C. `(3)/(5)`
D. 4

Answer

To solve the limit \( \lim_{x \to 2} \frac{x^3 + x^2 + 4x + 12}{x^3 - 3x + 2} \), we will follow these steps:

### Step 1: Substitute the limit value
We will substitute \( x = 2 \) directly into the function.

\[ L = \frac{2^3 + 2^2 + 4 \cdot 2 + 12}{2^3 - 3 \cdot 2 + 2} \]

### Step 2: Calculate the numerator
Now, we calculate the numerator:

\[ 2^3 = 8, \quad 2^2 = 4, \quad 4 \cdot 2 = 8 \]
So, the numerator becomes:

\[ 8 + 4 + 8 + 12 = 32 \]

### Step 3: Calculate the denominator
Next, we calculate the denominator:

\[ 2^3 = 8, \quad 3 \cdot 2 = 6 \]
So, the denominator becomes:

\[ 8 - 6 + 2 = 4 \]

### Step 4: Formulate the limit
Now we can substitute the calculated numerator and denominator back into the limit:

\[ L = \frac{32}{4} \]

### Step 5: Simplify the limit
Finally, we simplify the fraction:

\[ L = 8 \]

Thus, the limit is:

\[ \boxed{8} \]

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