Evaluate: `lim_(xrarr3) (x^(2)-4x+3)/(x^(2)+6x- 27)`

Options

A. `1/2`
B. `1/3`
C. `1/6`
D. `1`

Explanation

To evaluate the limit \( \lim_{x \to 3} \frac{x^2 - 4x + 3}{x^2 + 6x - 27} \), we will follow these steps:

### Step 1: Factor the numerator and denominator

First, we need to factor both the numerator and the denominator.

**Numerator:**
The expression \( x^2 - 4x + 3 \) can be factored as follows:
\[ x^2 - 4x + 3 = (x - 3)(x - 1) \]

**Denominator:**
The expression \( x^2 + 6x - 27 \) can be factored as follows:
\[ x^2 + 6x - 27 = (x + 9)(x - 3) \]

### Step 2: Rewrite the limit with the factored form

Now we can rewrite the limit using the factored forms:
\[ \lim_{x \to 3} \frac{(x - 3)(x - 1)}{(x + 9)(x - 3)} \]

### Step 3: Cancel the common factor

Notice that \( (x - 3) \) is a common factor in both the numerator and the denominator. We can cancel this factor:
\[ \lim_{x \to 3} \frac{(x - 1)}{(x + 9)} \]

### Step 4: Substitute \( x = 3 \)

Now we can directly substitute \( x = 3 \) into the simplified expression:
\[ \frac{(3 - 1)}{(3 + 9)} = \frac{2}{12} = \frac{1}{6} \]

### Final Result

Thus, the limit is:
\[ \lim_{x \to 3} \frac{x^2 - 4x + 3}{x^2 + 6x - 27} = \frac{1}{6} \]

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