Question

What is the largest possible value not in the domain of log (x-2)/(x^2-5)

Answer 1

To determine the largest possible value not in the domain of the function \( \log\left(\frac{x-2}{x^2-5}\right) \), we need to consider two conditions:

1. The argument of the logarithm must be positive:
\[
\frac{x-2}{x^2-5} > 0
\]

2. The denominator \( x^2 - 5 \) must not be zero:
\[
x^2 - 5 \neq 0 \quad \Rightarrow \quad x \neq \pm \sqrt{5}
\]

### Step 1: Determine where \( \frac{x-2}{x^2-5} > 0 \)

To find the intervals where the fraction is positive, we will analyze the signs of the numerator and denominator.

**Numerator: \( x - 2 \)**
- \( x - 2 > 0 \) when \( x > 2 \)
- \( x - 2 = 0 \) when \( x = 2 \)
- \( x - 2 < 0 \) when \( x < 2 \)

**Denominator: \( x^2 - 5 \)**
- \( x^2 - 5 > 0 \) when \( x < -\sqrt{5} \) or \( x > \sqrt{5} \)
- \( x^2 - 5 = 0 \) when \( x = \pm \sqrt{5} \)
- \( x^2 - 5 < 0 \) when \( -\sqrt{5} < x < \sqrt{5} \)

### Step 2: Analyze intervals

We identify the critical points:
- \( x = 2 \)
- \( x = -\sqrt{5} \) (approximately -2.236)
- \( x = \sqrt{5} \) (approximately 2.236)

This leads to the following intervals to test:
1. \( (-\infty, -\sqrt{5}) \)
2. \( (-\sqrt{5}, 2) \)
3. \( (2, \sqrt{5}) \)
4. \( (\sqrt{5}, \infty) \)

### Step 3: Test intervals

1. **For \( x < -\sqrt{5} \)** (e.g., \( x = -3 \)):
- Numerator: \( -3 - 2 < 0 \) (negative)
- Denominator: \( (-3)^2 - 5 = 9 - 5 > 0 \) (positive)
- Result: \( \frac{-}{+} < 0 \) (not in domain)

2. **For \( -\sqrt{5} < x < 2 \)** (e.g., \( x = 0 \)):
- Numerator: \( 0 - 2 < 0 \) (negative)
- Denominator: \( 0^2 - 5 < 0 \) (negative)
- Result: \( \frac{-}{-} > 0 \) (in domain)

3. **For \( 2 < x < \sqrt{5} \)** (e.g., \( x = 3 \)):
- Numerator: \( 3 - 2 > 0 \) (positive)
- Denominator: \( 3^2 - 5 = 9 - 5 > 0 \) (positive)
- Result: \( \frac{+}{+} > 0 \) (in domain)

4. **For \( x > \sqrt{5} \)** (e.g., \( x = 3 \)):
- Numerator: \( 3 - 2 > 0 \) (positive)
- Denominator: \( 3^2 - 5 = 9 - 5 > 0 \) (positive)
- Result: \( \frac{+}{+} > 0 \) (in domain)

### Step 4: Excluded values
- Excluded from the domain: \( x = -\sqrt{5} \), \( x = \sqrt{5} \), and \( x = 2 \).

### Step 5: Largest possible value not in the domain
The largest value of \( x \) not in the domain is:

\[
\sqrt{5} \quad (\text{approximately } 2.236)
\]

Thus, the answer is:

\[
\boxed{\sqrt{5}}
\]