To find the limit \( \lim_{{x \to 0}} \frac{\cos(1) - \cos(\cos(x))}{x^2} \), we can use L'Hospital's Rule, which is applicable because this limit results in the indeterminate form \( \frac{0}{0} \) as \( x \) approaches 0.

First, let's differentiate the numerator and the denominator with respect to \( x \) and then re-evaluate the limit.

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

### Step 1: Differentiate the Numerator
Let \( f(x) = \cos(1) - \cos(\cos(x)) \). Then the derivative \( f'(x) \) with respect to \( x \) is:

\[
f'(x) = 0 - \frac{d}{dx} \cos(\cos(x)) = \sin(\cos(x)) \cdot \sin(x) \cdot (-1)
\]

So,
\[
f'(x) = -\sin(\cos(x)) \cdot \sin(x)
\]

### Step 2: Differentiate the Denominator
Let \( g(x) = x^2 \). The derivative \( g'(x) \) with respect to \( x \) is:

\[
g'(x) = 2x
\]

### Step 3: Apply L'Hospital's Rule
Using L'Hospital's Rule, the limit becomes:

\[
\lim_{{x \to 0}} \frac{-\sin(\cos(x)) \cdot \sin(x)}{2x}
\]

### Step 4: Evaluate the limit again
This new limit as \( x \) tends to 0 still gives an indeterminate form, so we apply L'Hospital's Rule a second time, requiring us to differentiate the numerator and denominator again.

I'll calculate it now.The limit is given by:

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

Would you like more details or have any other questions?

Here are some related questions:
1. What is L'Hospital's Rule and when can it be applied?
2. How do you differentiate trigonometric functions?
3. What is the Taylor series expansion for cosine and sine functions?
4. Can you evaluate \(\sin(1)\) to a numerical value?
5. How do you find the limit of more complex trigonometric functions?
6. What is the significance of indeterminate forms in calculus?
7. How does the chain rule apply in differentiating composite functions?
8. What are common mistakes to avoid when applying L'Hospital's Rule?

**Tip:** Always verify that you are working with an indeterminate form before applying L'Hospital's Rule.

Learn how to evaluate the limit \( \lim_{x \to 0} \frac{\cos(1) - \cos(\cos(x))}{x^2} \) using L'Hospital's Rule. This problem involves limits, trigonometric functions, and the application of L'Hospital's Rule to resolve the indeterminate form \( \frac{0}{0} \). Suitable for advanced mathematics enthusiasts.

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