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The expression given is:

\[
\lim_{{x \to 0}} \left( \cos x \right)^{\frac{1}{x}}
\]

To solve this limit, let's start by rewriting the expression in a more convenient form for evaluating limits.

### Step 1: Use the logarithm transformation
We can apply the natural logarithm to simplify the power:

\[
y = \left( \cos x \right)^{\frac{1}{x}} \quad \Rightarrow \quad \ln y = \frac{1}{x} \ln (\cos x)
\]

Now, we will solve for \( \ln y \), and then exponentiate the result at the end.

### Step 2: Evaluate the limit of \( \frac{1}{x} \ln (\cos x) \)
We need to evaluate:

\[
\lim_{{x \to 0}} \frac{1}{x} \ln (\cos x)
\]

### Step 3: Apply series expansion or L'Hopital's rule
As \( x \to 0 \), \( \cos x \) approaches 1. We can use the approximation \( \cos x \approx 1 - \frac{x^2}{2} \) for small \( x \). Thus, \( \ln(\cos x) \approx -\frac{x^2}{2} \). Therefore:

\[
\frac{1}{x} \ln (\cos x) \approx \frac{1}{x} \left( -\frac{x^2}{2} \right) = -\frac{x}{2}
\]

As \( x \to 0 \), this expression tends to 0.

### Step 4: Exponentiate the result
Since \( \ln y \to 0 \), it follows that \( y = e^0 = 1 \).

Thus, the original limit is:

\[
\lim_{{x \to 0}} \left( \cos x \right)^{\frac{1}{x}} = 1
\]

Would you like more details or have any questions about this solution? Here are some related questions to further explore:

1. What happens if we change \( \cos x \) to another trigonometric function like \( \sin x \)?
2. How does L'Hopital's rule apply to other indeterminate forms?
3. Can we always use series expansions for small values of \( x \)?
4. How would the limit change if we considered \( \lim_{{x \to \infty}} \) instead of \( \lim_{{x \to 0}} \)?
5. How does the behavior of the cosine function affect the limit?

### Tip:
When working with limits involving exponents, using logarithmic transformations can often simplify the problem, especially when dealing with powers of functions approaching 0 or 1.

What is the limit of (cos(x))^(1/x) as x approaches 0?

Learn how to solve the limit lim_{x \to 0} (cos(x))^{1/x} using logarithmic transformations, Taylor series expansion, and L'Hopital's Rule. Step-by-step explanation for advanced calculus students.

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