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: Identify the Indeterminate Form
First, we evaluate the expression directly at \( x = 0 \):
\[ \cos(0) = 1 \quad \text{and} \quad \cot(0) = \frac{\cos(0)}{\sin(0)} = \frac{1}{0} \text{ (undefined)} \]
Thus, we have the form \( 1^{\infty} \), which is indeterminate.

**Hint:** When you encounter \( 1^{\infty} \), consider using logarithms to simplify the expression.

### Step 2: Take the Natural Logarithm
Let \( L = \lim_{x \to 0} (\cos x)^{\cot x} \). Taking the natural logarithm of both sides gives:
\[ \ln L = \lim_{x \to 0} \cot x \cdot \ln(\cos x) \]

**Hint:** Remember that \( \cot x = \frac{\cos x}{\sin x} \) can be useful for rewriting the limit.

### Step 3: Rewrite the Limit
Substituting \( \cot x \):
\[ \ln L = \lim_{x \to 0} \frac{\ln(\cos x)}{\tan x} \]

**Hint:** Use the Taylor series expansion for \( \cos x \) around \( x = 0 \) to simplify \( \ln(\cos x) \).

### Step 4: Use Taylor Series Expansion
The Taylor series for \( \cos x \) is:
\[ \cos x \approx 1 - \frac{x^2}{2} \quad \text{as } x \to 0 \]
Thus,
\[ \ln(\cos x) \approx \ln\left(1 - \frac{x^2}{2}\right) \approx -\frac{x^2}{2} \quad \text{(using } \ln(1 + u) \approx u \text{ for small } u\text{)} \]

**Hint:** Substitute this approximation back into the limit.

### Step 5: Substitute and Simplify
Now substituting back:
\[ \ln L = \lim_{x \to 0} \frac{-\frac{x^2}{2}}{\tan x} \]
Since \( \tan x \approx x \) as \( x \to 0 \):
\[ \ln L = \lim_{x \to 0} \frac{-\frac{x^2}{2}}{x} = \lim_{x \to 0} -\frac{x}{2} = 0 \]

**Hint:** If the limit evaluates to 0, it indicates that \( L \) approaches \( e^0 \).

### Step 6: Exponentiate to Find \( L \)
Since \( \ln L = 0 \):
\[ L = e^0 = 1 \]

### Final Answer
Thus, the value of \( \lim_{x \to 0} (\cos x)^{\cot x} \) is:
\[ \boxed{1} \]