The value of `lim_(xrarr0)(cosx)^(cotx)`, is

Options

A. e
B. `(1)/(e)`
C. `1`
D. `-1`

Explanation

To find the limit \( \lim_{x \to 0} (\cos x)^{\cot x} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the limit:
\[ \lim_{x \to 0} (\cos x)^{\cot x} \]
We know that \( \cot x = \frac{\cos x}{\sin x} \). Therefore, we can rewrite the limit as:
\[ \lim_{x \to 0} (\cos x)^{\frac{\cos x}{\sin x}} \]

### Step 2: Identify the form of the limit
As \( x \to 0 \), \( \cos x \to 1 \) and \( \cot x \to \infty \). This gives us an indeterminate form of \( 1^{\infty} \).

### Step 3: Use the exponential limit property
We can use the property that if \( \lim_{x \to a} f(x)^{g(x)} \) results in \( 1^{\infty} \), we can express it as:
\[ \lim_{x \to a} f(x)^{g(x)} = e^{\lim_{x \to a} (f(x) - 1) \cdot g(x)} \]
In our case, let \( f(x) = \cos x \) and \( g(x) = \cot x \). Thus, we rewrite the limit:
\[ \lim_{x \to 0} (\cos x)^{\cot x} = e^{\lim_{x \to 0} (\cos x - 1) \cdot \cot x} \]

### Step 4: Simplify the expression
Now we need to evaluate:
\[ \lim_{x \to 0} (\cos x - 1) \cdot \cot x \]
We know that \( \cos x - 1 \) can be approximated using the Taylor series expansion:
\[ \cos x \approx 1 - \frac{x^2}{2} \quad \text{as } x \to 0 \]
Thus:
\[ \cos x - 1 \approx -\frac{x^2}{2} \]
And since \( \cot x = \frac{\cos x}{\sin x} \) and \( \sin x \approx x \) as \( x \to 0 \), we have:
\[ \cot x \approx \frac{1}{x} \]

### Step 5: Substitute and find the limit
Now substituting these approximations:
\[ \lim_{x \to 0} \left(-\frac{x^2}{2}\right) \cdot \frac{1}{x} = \lim_{x \to 0} -\frac{x}{2} = 0 \]

### Step 6: Final result
Thus, we have:
\[ \lim_{x \to 0} (\cos x)^{\cot x} = e^{0} = 1 \]

### Conclusion
The value of \( \lim_{x \to 0} (\cos x)^{\cot x} \) is:
\[ \boxed{1} \]