The value of ` lim_(xrarr 0) (1-cos(1-cos x))/(x^4)` is equal to

Options

A. `(1)(8)`
B. `(1)/(2)`
C. `(1)/(4)`
D. none of these

Answer

To solve the limit \( \lim_{x \to 0} \frac{1 - \cos(1 - \cos x)}{x^4} \), we will follow these steps:

### Step 1: Identify the limit form
First, we need to evaluate the limit as \( x \) approaches 0. We notice that both the numerator and denominator approach 0, which indicates that we have an indeterminate form of type \( \frac{0}{0} \).

### Step 2: Use the identity for cosine
We can use the trigonometric identity for \( 1 - \cos y \):
\[ 1 - \cos y \approx \frac{y^2}{2} \quad \text{as } y \to 0. \]
Here, we will set \( y = 1 - \cos x \).

### Step 3: Find \( 1 - \cos x \) as \( x \to 0 \)
Using the same identity, we find:
\[ 1 - \cos x \approx \frac{x^2}{2} \quad \text{as } x \to 0. \]
Thus, as \( x \to 0 \):
\[ y = 1 - \cos x \approx \frac{x^2}{2}. \]

### Step 4: Substitute \( y \) into the limit
Now substituting \( y \) into our limit:
\[ \lim_{x \to 0} \frac{1 - \cos(1 - \cos x)}{x^4} = \lim_{x \to 0} \frac{1 - \cos\left(\frac{x^2}{2}\right)}{x^4}. \]

### Step 5: Apply the cosine identity again
Now, we apply the cosine identity again:
\[ 1 - \cos\left(\frac{x^2}{2}\right) \approx \frac{\left(\frac{x^2}{2}\right)^2}{2} = \frac{x^4}{8} \quad \text{as } x \to 0. \]

### Step 6: Substitute back into the limit
Now substituting this back into our limit:
\[ \lim_{x \to 0} \frac{\frac{x^4}{8}}{x^4} = \lim_{x \to 0} \frac{1}{8} = \frac{1}{8}. \]

### Final Answer
Thus, the value of the limit is:
\[ \boxed{\frac{1}{8}}. \]