Evaluate: `lim_(xrarr2) (x^(3)-2x^(2))/(x^(2)-5x+6)`

Options

A. `4`
B. `-4`
C. `-1/4`
D. `1/4`

Explanation

To evaluate the limit \( \lim_{x \to 2} \frac{x^3 - 2x^2}{x^2 - 5x + 6} \), we will follow these steps:

### Step 1: Substitute \( x = 2 \)

First, we will substitute \( x = 2 \) directly into the expression to check if we can evaluate the limit directly.

\[ \text{Numerator: } 2^3 - 2 \cdot 2^2 = 8 - 8 = 0 \]
\[ \text{Denominator: } 2^2 - 5 \cdot 2 + 6 = 4 - 10 + 6 = 0 \]

Since both the numerator and denominator evaluate to 0, we have an indeterminate form \( \frac{0}{0} \). We need to simplify the expression.

### Step 2: Factor the numerator and denominator

Next, we will factor both the numerator and the denominator.

**Numerator:**
\[ x^3 - 2x^2 = x^2(x - 2) \]

**Denominator:**
To factor the denominator \( x^2 - 5x + 6 \), we look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3.
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 3: Rewrite the limit

Now we can rewrite the limit using the factored forms:

\[ \lim_{x \to 2} \frac{x^2(x - 2)}{(x - 2)(x - 3)} \]

### Step 4: Cancel common factors

We can cancel the common factor \( (x - 2) \) from the numerator and the denominator (valid since we are taking the limit as \( x \) approaches 2, not equal to 2):

\[ \lim_{x \to 2} \frac{x^2}{x - 3} \]

### Step 5: Substitute \( x = 2 \) again

Now we substitute \( x = 2 \) into the simplified expression:

\[ \frac{2^2}{2 - 3} = \frac{4}{-1} = -4 \]

### Final Answer

Thus, the limit is:

\[ \lim_{x \to 2} \frac{x^3 - 2x^2}{x^2 - 5x + 6} = -4 \]

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