What is `lim_(thetato0)(sqrt(1-costheta))/(theta)` equal to?

Options

A. `sqrt2`
B. `2sqrt2`
C. `(1)/(sqrt2)`
D. `-(1)/(2sqrt2)`

Explanation

To solve the limit \( \lim_{\theta \to 0} \frac{\sqrt{1 - \cos \theta}}{\theta} \), we can follow these steps:

### Step 1: Rewrite \(1 - \cos \theta\)
We can use the trigonometric identity:
\[ 1 - \cos \theta = 2 \sin^2\left(\frac{\theta}{2}\right) \]
Thus, we can rewrite the limit as:
\[ \lim_{\theta \to 0} \frac{\sqrt{1 - \cos \theta}}{\theta} = \lim_{\theta \to 0} \frac{\sqrt{2 \sin^2\left(\frac{\theta}{2}\right)}}{\theta} \]

### Step 2: Simplify the Square Root
The square root can be simplified:
\[ \sqrt{2 \sin^2\left(\frac{\theta}{2}\right)} = \sqrt{2} \cdot \sin\left(\frac{\theta}{2}\right) \]
Now, the limit becomes:
\[ \lim_{\theta \to 0} \frac{\sqrt{2} \cdot \sin\left(\frac{\theta}{2}\right)}{\theta} \]

### Step 3: Adjust the Limit
To make it easier to compute the limit, we can multiply and divide by \( \frac{1}{2} \):
\[ \lim_{\theta \to 0} \frac{\sqrt{2} \cdot \sin\left(\frac{\theta}{2}\right)}{\theta} = \lim_{\theta \to 0} \frac{\sqrt{2} \cdot \sin\left(\frac{\theta}{2}\right)}{\frac{\theta}{2}} \cdot \frac{1}{2} \]

### Step 4: Use the Limit Property
We know that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
So, substituting \( x = \frac{\theta}{2} \), we have:
\[ \lim_{\theta \to 0} \frac{\sin\left(\frac{\theta}{2}\right)}{\frac{\theta}{2}} = 1 \]

### Step 5: Final Calculation
Now we can evaluate the limit:
\[ \lim_{\theta \to 0} \frac{\sqrt{2} \cdot \sin\left(\frac{\theta}{2}\right)}{\theta} = \sqrt{2} \cdot 1 \cdot \frac{1}{2} = \frac{\sqrt{2}}{2} \]

### Conclusion
Thus, the limit is:
\[ \lim_{\theta \to 0} \frac{\sqrt{1 - \cos \theta}}{\theta} = \frac{1}{\sqrt{2}} \]