Evaluate `lim_(xto0) (sqrt(2)-sqrt(1+cosx))/(sin^(2)x).`

Explanation

To evaluate the limit \( \lim_{x \to 0} \frac{\sqrt{2} - \sqrt{1 + \cos x}}{\sin^2 x} \), we will follow these steps:

### Step 1: Rationalize the Numerator
We start by rationalizing the numerator. We multiply the numerator and the denominator by the conjugate of the numerator, which is \( \sqrt{2} + \sqrt{1 + \cos x} \):

\[ \lim_{x \to 0} \frac{\sqrt{2} - \sqrt{1 + \cos x}}{\sin^2 x} \cdot \frac{\sqrt{2} + \sqrt{1 + \cos x}}{\sqrt{2} + \sqrt{1 + \cos x}} = \lim_{x \to 0} \frac{(\sqrt{2})^2 - (\sqrt{1 + \cos x})^2}{\sin^2 x (\sqrt{2} + \sqrt{1 + \cos x})} \]

### Step 2: Simplify the Numerator
The numerator simplifies to:

\[ 2 - (1 + \cos x) = 2 - 1 - \cos x = 1 - \cos x \]

Thus, we can rewrite our limit as:

\[ \lim_{x \to 0} \frac{1 - \cos x}{\sin^2 x (\sqrt{2} + \sqrt{1 + \cos x})} \]

### Step 3: Use the Identity for \(1 - \cos x\)
We can use the trigonometric identity \( 1 - \cos x = 2 \sin^2 \left(\frac{x}{2}\right) \):

\[ \lim_{x \to 0} \frac{2 \sin^2 \left(\frac{x}{2}\right)}{\sin^2 x (\sqrt{2} + \sqrt{1 + \cos x})} \]

### Step 4: Rewrite \(\sin^2 x\)
Using the identity \( \sin x = 2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \), we have:

\[ \sin^2 x = 4 \sin^2 \left(\frac{x}{2}\right) \cos^2 \left(\frac{x}{2}\right) \]

Substituting this into our limit gives:

\[ \lim_{x \to 0} \frac{2 \sin^2 \left(\frac{x}{2}\right)}{4 \sin^2 \left(\frac{x}{2}\right) \cos^2 \left(\frac{x}{2}\right) (\sqrt{2} + \sqrt{1 + \cos x})} \]

### Step 5: Cancel \(\sin^2 \left(\frac{x}{2}\right)\)
We can cancel \( \sin^2 \left(\frac{x}{2}\right) \) from the numerator and the denominator:

\[ \lim_{x \to 0} \frac{2}{4 \cos^2 \left(\frac{x}{2}\right) (\sqrt{2} + \sqrt{1 + \cos x})} \]

### Step 6: Evaluate the Limit
As \( x \to 0 \), \( \cos \left(\frac{x}{2}\right) \to 1 \) and \( \sqrt{1 + \cos x} \to \sqrt{2} \):

\[ \lim_{x \to 0} \frac{2}{4 \cdot 1^2 \cdot (\sqrt{2} + \sqrt{2})} = \frac{2}{4 \cdot 2\sqrt{2}} = \frac{2}{8\sqrt{2}} = \frac{1}{4\sqrt{2}} \]

### Final Answer
Thus, the limit is:

\[ \lim_{x \to 0} \frac{\sqrt{2} - \sqrt{1 + \cos x}}{\sin^2 x} = \frac{1}{4\sqrt{2}} \]