The value of `lim_(xto0) (1-(cosx)sqrt(cos2x))/(x^(2))` is

Options

A. `(1)/(3)`
B. `(1)/(2)`
C. `(1)/(6)`
D. `(5)/(12)`

Explanation

To find the limit

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

we start by substituting \( x = 0 \):

1. **Substitution**:
\[ \cos(0) = 1 \quad \text{and} \quad \cos(2 \cdot 0) = 1. \]
Therefore, the expression becomes:
\[ \frac{1 - 1 \cdot \sqrt{1}}{0^2} = \frac{0}{0}, \]
which is an indeterminate form.

**Hint**: If you encounter a \( \frac{0}{0} \) form, consider using algebraic manipulation or L'Hôpital's rule.

2. **Rationalization**: To resolve the indeterminate form, we can rationalize the numerator. We multiply and divide by the conjugate:
\[ \frac{(1 - \cos x \sqrt{\cos 2x})(1 + \cos x \sqrt{\cos 2x})}{(1 + \cos x \sqrt{\cos 2x})}. \]
This gives us:
\[ \frac{1 - (\cos x \sqrt{\cos 2x})^2}{x^2(1 + \cos x \sqrt{\cos 2x})}. \]

**Hint**: Rationalizing helps to eliminate the square root and simplify the expression.

3. **Simplifying the Numerator**: We can simplify the numerator:
\[ 1 - \cos^2 x \cos 2x = \sin^2 x \cos 2x. \]
Thus, our limit now looks like:
\[ \lim_{x \to 0} \frac{\sin^2 x \cos 2x}{x^2(1 + \cos x \sqrt{\cos 2x})}. \]

**Hint**: Use trigonometric identities to simplify expressions involving sine and cosine.

4. **Applying Limit**: Now we can apply the limit:
\[ \lim_{x \to 0} \frac{\sin^2 x}{x^2} \cdot \lim_{x \to 0} \frac{\cos 2x}{1 + \cos x \sqrt{\cos 2x}}. \]
We know that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \implies \lim_{x \to 0} \frac{\sin^2 x}{x^2} = 1. \]
Also, as \( x \to 0 \):
\[ \cos 2x \to 1 \quad \text{and} \quad \cos x \to 1 \implies \sqrt{\cos 2x} \to 1. \]
Therefore,
\[ 1 + \cos x \sqrt{\cos 2x} \to 2. \]

**Hint**: Break down the limit into manageable parts, and evaluate each part separately.

5. **Final Calculation**: Now we can combine these results:
\[ \lim_{x \to 0} \frac{\sin^2 x}{x^2} \cdot \frac{1}{2} = 1 \cdot \frac{1}{2} = \frac{1}{2}. \]

Thus, the value of the limit is

\[ \boxed{\frac{1}{2}}. \]