Evaluate the following limits:
`lim_(xto pi)(sqrt(2+cosx)-1)/((pi-x)^(2))`

Options

A. `1/4`
B. `1/2`
C. `2`
D. Doesn't exist

Answer

To evaluate the limit

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

we will follow these steps:

### Step 1: Substitute \( x = \pi \)

First, we substitute \( x = \pi \) into the expression:

\[ \sqrt{2 + \cos(\pi)} - 1 = \sqrt{2 - 1} - 1 = \sqrt{1} - 1 = 0, \]

and

\[ (\pi - \pi)^2 = 0. \]

This gives us the indeterminate form \( \frac{0}{0} \).

### Step 2: Apply L'Hôpital's Rule

Since we have the \( \frac{0}{0} \) form, we can apply L'Hôpital's Rule. We differentiate the numerator and the denominator:

- **Numerator**: The derivative of \( \sqrt{2 + \cos x} - 1 \) is

\[ \frac{1}{2\sqrt{2 + \cos x}} \cdot (-\sin x) = -\frac{\sin x}{2\sqrt{2 + \cos x}}. \]

- **Denominator**: The derivative of \( (\pi - x)^2 \) is

\[ 2(\pi - x)(-1) = -2(\pi - x). \]

### Step 3: Rewrite the limit

Now we rewrite the limit using these derivatives:

\[ \lim_{x \to \pi} \frac{-\frac{\sin x}{2\sqrt{2 + \cos x}}}{-2(\pi - x)} = \lim_{x \to \pi} \frac{\sin x}{4(\pi - x)\sqrt{2 + \cos x}}. \]

### Step 4: Substitute \( x = \pi \) again

Now we substitute \( x = \pi \):

\[ \sin(\pi) = 0, \]

and

\[ \sqrt{2 + \cos(\pi)} = \sqrt{2 - 1} = 1. \]

Thus, we have another \( \frac{0}{0} \) form. We apply L'Hôpital's Rule again.

### Step 5: Differentiate again

We differentiate the numerator and denominator again:

- **Numerator**: The derivative of \( \sin x \) is \( \cos x \).
- **Denominator**: Using the product rule on \( 4(\pi - x)\sqrt{2 + \cos x} \):

\[ 4\left(-\sqrt{2 + \cos x} + (\pi - x) \cdot \frac{-\sin x}{2\sqrt{2 + \cos x}}\right). \]

### Step 6: Rewrite the limit again

Now we have:

\[ \lim_{x \to \pi} \frac{\cos x}{4\left(-\sqrt{2 + \cos x} + (\pi - x) \cdot \frac{-\sin x}{2\sqrt{2 + \cos x}}\right)}. \]

### Step 7: Substitute \( x = \pi \) again

Substituting \( x = \pi \):

\[ \cos(\pi) = -1, \]

and

\[ \sqrt{2 + \cos(\pi)} = 1. \]

Thus, we have:

\[ 4\left(-1 + 0\right) = -4. \]

### Step 8: Final limit evaluation

Putting it all together:

\[ \lim_{x \to \pi} \frac{-1}{-4} = \frac{1}{4}. \]

### Conclusion

Thus, the limit is

\[ \frac{1}{4}. \]