Evaluate `lim_(xto0) (8)/(x^(8)){1-"cos"(x^(2))/(2)-"cos"(x^(2))/(4)+"cos"(x^(2))/(2)"cos"(x^(2))/(4)}.`

Explanation

To evaluate the limit

\[ \lim_{x \to 0} \frac{8}{x^8} \left( 1 - \cos\left(\frac{x^2}{2}\right) - \cos\left(\frac{x^2}{4}\right) + \cos\left(\frac{x^2}{2}\right) \cos\left(\frac{x^2}{4}\right) \right), \]

we will follow these steps:

### Step 1: Rewrite the expression
The expression can be rewritten as:

\[ \lim_{x \to 0} \frac{8}{x^8} \left( \left(1 - \cos\left(\frac{x^2}{2}\right)\right) + \left(1 - \cos\left(\frac{x^2}{4}\right)\right) - \left(1 - \cos\left(\frac{x^2}{2}\right)\right)\left(1 - \cos\left(\frac{x^2}{4}\right)\right) \right). \]

### Step 2: Use the identity for cosine
Using the identity \(1 - \cos(a) = 2\sin^2\left(\frac{a}{2}\right)\), we can express:

\[ 1 - \cos\left(\frac{x^2}{2}\right) = 2\sin^2\left(\frac{x^2}{4}\right), \]
\[ 1 - \cos\left(\frac{x^2}{4}\right) = 2\sin^2\left(\frac{x^2}{8}\right). \]

Substituting these into our limit gives:

\[ \lim_{x \to 0} \frac{8}{x^8} \left( 2\sin^2\left(\frac{x^2}{4}\right) + 2\sin^2\left(\frac{x^2}{8}\right) - 4\sin^2\left(\frac{x^2}{4}\right)\sin^2\left(\frac{x^2}{8}\right) \right). \]

### Step 3: Factor out constants
We can factor out the constants:

\[ = \lim_{x \to 0} \frac{8}{x^8} \cdot 2 \left( \sin^2\left(\frac{x^2}{4}\right) + \sin^2\left(\frac{x^2}{8}\right) - 2\sin^2\left(\frac{x^2}{4}\right)\sin^2\left(\frac{x^2}{8}\right) \right). \]

### Step 4: Simplify the limit
Now we can simplify the limit. As \(x \to 0\), we can use the small angle approximation \(\sin(x) \approx x\):

\[ \sin\left(\frac{x^2}{4}\right) \approx \frac{x^2}{4}, \quad \sin\left(\frac{x^2}{8}\right) \approx \frac{x^2}{8}. \]

Thus,

\[ \sin^2\left(\frac{x^2}{4}\right) \approx \left(\frac{x^2}{4}\right)^2 = \frac{x^4}{16}, \quad \sin^2\left(\frac{x^2}{8}\right) \approx \left(\frac{x^2}{8}\right)^2 = \frac{x^4}{64}. \]

### Step 5: Substitute back into the limit
Substituting these approximations back into the limit gives:

\[ \lim_{x \to 0} \frac{8}{x^8} \cdot 2 \left( \frac{x^4}{16} + \frac{x^4}{64} - 2 \cdot \frac{x^4}{16} \cdot \frac{x^4}{64} \right). \]

### Step 6: Combine terms
Combining the terms inside the limit:

\[ = \lim_{x \to 0} \frac{8}{x^8} \cdot 2 \left( \frac{x^4}{16} + \frac{x^4}{64} - \frac{x^8}{1024} \right). \]

### Step 7: Factor out \(x^4\)
Factoring out \(x^4\):

\[ = \lim_{x \to 0} \frac{8 \cdot 2}{x^8} \cdot x^4 \left( \frac{1}{16} + \frac{1}{64} - \frac{x^4}{1024} \right). \]

### Step 8: Evaluate the limit
As \(x \to 0\), the term \(-\frac{x^4}{1024}\) approaches 0:

\[ = \lim_{x \to 0} \frac{16}{x^8} \cdot x^4 \left( \frac{1}{16} + \frac{1}{64} \right) = \lim_{x \to 0} \frac{16}{x^8} \cdot x^4 \cdot \frac{5}{64} = \lim_{x \to 0} \frac{80}{64 x^4} = \frac{80}{64} \cdot \lim_{x \to 0} \frac{1}{x^4}. \]

### Final Answer
Thus, the limit evaluates to:

\[ \frac{80}{64} = \frac{5}{4}. \]