`lim_(x to 0)(sqrt((1/2)(1- cos 2x)))/x=`

Options

A. 1
B. `-1`
C. 0
D. none

Answer

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

### Step 1: Simplify the expression inside the limit
We start with the expression:
\[ \lim_{x \to 0} \frac{\sqrt{\frac{1}{2}(1 - \cos 2x)}}{x} \]

### Step 2: Use the trigonometric identity for \(1 - \cos 2x\)
Recall the identity:
\[ 1 - \cos 2x = 2 \sin^2 x \]
Using this identity, we can rewrite the limit:
\[ \lim_{x \to 0} \frac{\sqrt{\frac{1}{2}(2 \sin^2 x)}}{x} \]

### Step 3: Simplify the square root
The expression simplifies to:
\[ \lim_{x \to 0} \frac{\sqrt{\sin^2 x}}{x} \]
Since \(\sqrt{\sin^2 x} = |\sin x|\), and as \(x\) approaches 0, \(\sin x\) is positive, we can drop the absolute value:
\[ \lim_{x \to 0} \frac{\sin x}{x} \]

### Step 4: Evaluate the limit
We know from the standard limit result that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

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