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

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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:

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