STATEMENT -1 : `f(x)= (1 -cos(1-cosx))/x^(4)` is continuous if f(0)=1/8
and
STATEMENT -2 : ` lim_( x to 0^(+)) f(x) = lim_(x to 0^(-)) f(x) = 1/8`

Options

A. Statement -1 is True, Statement -2 is True, Statement -2 is a correct explantion for Statement -1
B. Statement -1 is True, Statement -2 is True, Statement -2 is NOT a correct explanation for Statement -1
C. Statement -1 is True , Statement -2 is Flase
D. Statement -1 is Flase , Statement - 2 is True

Answer

To determine the continuity of the function \( f(x) = \frac{1 - \cos(1 - \cos x)}{x^4} \) at \( x = 0 \) and to evaluate the limits as \( x \) approaches 0 from both sides, we will follow these steps:

### Step 1: Evaluate \( f(0) \)
We need to define \( f(0) \) to check continuity. The statement claims that \( f(0) = \frac{1}{8} \).

### Step 2: Find the limit as \( x \) approaches 0
We need to compute:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{1 - \cos(1 - \cos x)}{x^4} \]

### Step 3: Analyze the limit
As \( x \to 0 \):
- \( \cos x \to 1 \)
- Therefore, \( 1 - \cos x \to 0 \)
- Consequently, \( 1 - \cos(1 - \cos x) \to 1 - \cos(0) = 0 \)

Both the numerator and denominator approach 0, which indicates we can apply L'Hôpital's Rule or use Taylor series expansion.

### Step 4: Use Taylor series expansion
Using the Taylor series expansion for \( \cos x \):
\[ \cos x \approx 1 - \frac{x^2}{2} + O(x^4) \]
Thus,
\[ 1 - \cos x \approx \frac{x^2}{2} \]
Substituting this into \( f(x) \):
\[ 1 - \cos(1 - \cos x) \approx 1 - \cos\left(\frac{x^2}{2}\right) \]
Using the Taylor expansion for \( \cos \):
\[ \cos\left(\frac{x^2}{2}\right) \approx 1 - \frac{(x^2/2)^2}{2} = 1 - \frac{x^4}{8} \]
So,
\[ 1 - \cos\left(\frac{x^2}{2}\right) \approx \frac{x^4}{8} \]

### Step 5: Substitute back into the limit
Now substituting back into the limit:
\[ f(x) \approx \frac{\frac{x^4}{8}}{x^4} = \frac{1}{8} \]
Thus,
\[ \lim_{x \to 0} f(x) = \frac{1}{8} \]

### Step 6: Check continuity
Since:
\[ \lim_{x \to 0} f(x) = \frac{1}{8} = f(0) \]
we conclude that \( f(x) \) is continuous at \( x = 0 \).

### Conclusion
Both statements are correct:
- **Statement 1**: \( f(x) \) is continuous at \( x = 0 \) if \( f(0) = \frac{1}{8} \).
- **Statement 2**: \( \lim_{x \to 0^{+}} f(x) = \lim_{x \to 0^{-}} f(x) = \frac{1}{8} \).