The value of `underset(theta to -pi/4)lim (cos theta +sin theta)/(theta+pi/4)` is

Options

A. `pi/4`
B. `(-pi)/(4)`
C. `-sqrt2`
D. `sqrt2`

Answer

To solve the limit \( \lim_{\theta \to -\frac{\pi}{4}} \frac{\cos \theta + \sin \theta}{\theta + \frac{\pi}{4}} \), we will follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( \theta = -\frac{\pi}{4} \) into the expression:

\[ \cos\left(-\frac{\pi}{4}\right) + \sin\left(-\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 0 \]

And for the denominator:

\[ -\frac{\pi}{4} + \frac{\pi}{4} = 0 \]

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

### Step 2: Apply L'Hôpital's Rule
Since we have the indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which states that we can take the derivative of the numerator and the denominator.

The numerator is \( \cos \theta + \sin \theta \) and the denominator is \( \theta + \frac{\pi}{4} \).

- Derivative of the numerator:
\[ \frac{d}{d\theta}(\cos \theta + \sin \theta) = -\sin \theta + \cos \theta \]

- Derivative of the denominator:
\[ \frac{d}{d\theta}(\theta + \frac{\pi}{4}) = 1 \]

### Step 3: Rewrite the limit
Now we can rewrite the limit using the derivatives:

\[ \lim_{\theta \to -\frac{\pi}{4}} \frac{-\sin \theta + \cos \theta}{1} \]

### Step 4: Substitute the limit value again
Now substitute \( \theta = -\frac{\pi}{4} \):

\[ -\sin\left(-\frac{\pi}{4}\right) + \cos\left(-\frac{\pi}{4}\right) = -\left(-\frac{1}{\sqrt{2}}\right) + \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Final Answer
Thus, the value of the limit is:

\[ \sqrt{2} \]