Evaluate `lim_(x to 2) [(1)/(x - 2) - (2(2x - 3))/(x^(3) - 3x^(2) + 2x)]`

Explanation

To evaluate the limit

\[ \lim_{x \to 2} \left[ \frac{1}{x - 2} - \frac{2(2x - 3)}{x^3 - 3x^2 + 2x} \right], \]

we will follow these steps:

### Step 1: Simplify the Denominator
First, we simplify the denominator \(x^3 - 3x^2 + 2x\). We can factor out \(x\):

\[ x^3 - 3x^2 + 2x = x(x^2 - 3x + 2). \]

Next, we can factor the quadratic \(x^2 - 3x + 2\):

\[ x^2 - 3x + 2 = (x - 1)(x - 2). \]

So, the complete factorization of the denominator is:

\[ x^3 - 3x^2 + 2x = x(x - 1)(x - 2). \]

### Step 2: Rewrite the Expression
Now we can rewrite the limit expression:

\[ \lim_{x \to 2} \left[ \frac{1}{x - 2} - \frac{2(2x - 3)}{x(x - 1)(x - 2)} \right]. \]

### Step 3: Combine the Fractions
To combine the two fractions, we need a common denominator, which is \(x(x - 1)(x - 2)\):

\[ \lim_{x \to 2} \left[ \frac{x(x - 1) - 2(2x - 3)}{x(x - 1)(x - 2)} \right]. \]

### Step 4: Simplify the Numerator
Now, simplify the numerator:

\[ x(x - 1) - 2(2x - 3) = x^2 - x - (4x - 6) = x^2 - x - 4x + 6 = x^2 - 5x + 6. \]

### Step 5: Factor the Numerator
Next, we factor \(x^2 - 5x + 6\):

\[ x^2 - 5x + 6 = (x - 2)(x - 3). \]

### Step 6: Substitute Back into the Limit
Now we substitute this back into our limit:

\[ \lim_{x \to 2} \frac{(x - 2)(x - 3)}{x(x - 1)(x - 2)}. \]

### Step 7: Cancel Common Factors
We can cancel the \((x - 2)\) terms:

\[ \lim_{x \to 2} \frac{x - 3}{x(x - 1)}. \]

### Step 8: Evaluate the Limit
Now we can directly substitute \(x = 2\):

\[ \frac{2 - 3}{2(2 - 1)} = \frac{-1}{2 \cdot 1} = -\frac{1}{2}. \]

Thus, the limit is

\[ \boxed{-\frac{1}{2}}. \]