The value of `lim_(xrarr0) (sqrt(1-cosx^(2)))/(1-cos x)` is

Options

A. `(1)/(2)`
B. `2`
C. `sqrt2`
D. none of these

Explanation

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

### Step 1: Rewrite the expressions using trigonometric identities
We know that \( 1 - \cos \theta = 2 \sin^2\left(\frac{\theta}{2}\right) \). Using this identity, we can rewrite both the numerator and the denominator.

\[ 1 - \cos(x^2) = 2 \sin^2\left(\frac{x^2}{2}\right) \]
\[ 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \]

### Step 2: Substitute these identities into the limit
Now, substituting these into our limit gives us:

\[ \lim_{x \to 0} \frac{\sqrt{2 \sin^2\left(\frac{x^2}{2}\right)}}{2 \sin^2\left(\frac{x}{2}\right)} \]

### Step 3: Simplify the expression
The square root in the numerator simplifies:

\[ = \lim_{x \to 0} \frac{\sqrt{2} \sin\left(\frac{x^2}{2}\right)}{2 \sin^2\left(\frac{x}{2}\right)} \]

### Step 4: Factor out constants
We can factor out \( \frac{\sqrt{2}}{2} \):

\[ = \frac{\sqrt{2}}{2} \lim_{x \to 0} \frac{\sin\left(\frac{x^2}{2}\right)}{\sin^2\left(\frac{x}{2}\right)} \]

### Step 5: Change variables for the limit
Let \( y = \frac{x}{2} \), then as \( x \to 0 \), \( y \to 0 \) as well. Thus, \( x^2 = 4y^2 \) and we can rewrite the limit:

\[ = \frac{\sqrt{2}}{2} \lim_{y \to 0} \frac{\sin(2y^2)}{\sin^2(y)} \]

### Step 6: Apply the limit
Using the small angle approximation \( \sin z \approx z \) as \( z \to 0 \):

\[ = \frac{\sqrt{2}}{2} \lim_{y \to 0} \frac{2y^2}{y^2} = \frac{\sqrt{2}}{2} \cdot 2 = \sqrt{2} \]

### Final Result
Thus, the limit is:

\[ \lim_{x \to 0} \frac{\sqrt{1 - \cos(x^2)}}{1 - \cos x} = \sqrt{2} \]

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