`underset(x to sqrt(2))(It) (x^(2) - 2)/(x^(2) + sqrt(2)x - 4)` is equal to

Options

A. `(3)/(2)`
B. `(1)/(2)`
C. `(2)/(3)`
D. 2

Answer

To solve the limit problem \( \lim_{x \to \sqrt{2}} \frac{x^2 - 2}{x^2 + \sqrt{2}x - 4} \), we will follow these steps:

### Step 1: Substitute \( x = \sqrt{2} \)

First, we substitute \( x = \sqrt{2} \) into the expression to check if we get a determinate form.

\[ \text{Numerator: } (\sqrt{2})^2 - 2 = 2 - 2 = 0 \]
\[ \text{Denominator: } (\sqrt{2})^2 + \sqrt{2}(\sqrt{2}) - 4 = 2 + 2 - 4 = 0 \]

Since both the numerator and denominator approach 0, we have an indeterminate form \( \frac{0}{0} \).

**Hint:** When you get an indeterminate form, you need to factor or simplify the expression.

### Step 2: Factor the Numerator and Denominator

Next, we factor the numerator and the denominator.

**Numerator:**
\[ x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2}) \]

**Denominator:**
To factor the denominator \( x^2 + \sqrt{2}x - 4 \), we need to find its roots. We know one root is \( \sqrt{2} \). The product of the roots (using Vieta's formulas) gives us:
\[ \text{Let the other root be } r. \quad \sqrt{2} \cdot r = -4 \implies r = -\frac{4}{\sqrt{2}} = -2\sqrt{2} \]
Thus, we can factor the denominator as:
\[ x^2 + \sqrt{2}x - 4 = (x - \sqrt{2})(x + 2\sqrt{2}) \]

**Hint:** Use Vieta's formulas to find the other root when one root is known.

### Step 3: Rewrite the Limit

Now we can rewrite the limit with the factored forms:

\[ \lim_{x \to \sqrt{2}} \frac{(x - \sqrt{2})(x + \sqrt{2})}{(x - \sqrt{2})(x + 2\sqrt{2})} \]

**Hint:** Cancel out the common factors in the numerator and denominator.

### Step 4: Cancel Common Factors

We can cancel \( (x - \sqrt{2}) \) from the numerator and denominator:

\[ \lim_{x \to \sqrt{2}} \frac{x + \sqrt{2}}{x + 2\sqrt{2}} \]

### Step 5: Substitute \( x = \sqrt{2} \) Again

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

\[ \frac{\sqrt{2} + \sqrt{2}}{\sqrt{2} + 2\sqrt{2}} = \frac{2\sqrt{2}}{3\sqrt{2}} = \frac{2}{3} \]

### Final Answer

Thus, the limit is:

\[ \boxed{\frac{2}{3}} \]