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

Options

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

Answer

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

### Step 1: Use the trigonometric identity
We know from trigonometric identities that:
\[ 1 - \cos \theta = 2 \sin^2\left(\frac{\theta}{2}\right) \]
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 expression
This simplifies to:
\[ \lim_{\theta \to 0} \frac{\sqrt{2} \sin\left(\frac{\theta}{2}\right)}{\theta} \]

### Step 3: Rewrite \(\theta\) in terms of \(\frac{\theta}{2}\)
Notice that:
\[ \theta = 2 \cdot \frac{\theta}{2} \]
So we can rewrite the limit as:
\[ \lim_{\theta \to 0} \frac{\sqrt{2} \sin\left(\frac{\theta}{2}\right)}{2 \cdot \frac{\theta}{2}} = \frac{\sqrt{2}}{2} \lim_{\theta \to 0} \frac{\sin\left(\frac{\theta}{2}\right)}{\frac{\theta}{2}} \]

### Step 4: Apply the limit property
We know that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
Thus, as \(\frac{\theta}{2} \to 0\), we have:
\[ \lim_{\theta \to 0} \frac{\sin\left(\frac{\theta}{2}\right)}{\frac{\theta}{2}} = 1 \]

### Step 5: Combine the results
Now substituting this back into our expression gives:
\[ \frac{\sqrt{2}}{2} \cdot 1 = \frac{\sqrt{2}}{2} \]

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