`lim_(x to 0) (log cos x)/x` is equal to:

Options

A. 1
B. 0
C. e
D. none of these

Answer

To solve the limit \( \lim_{x \to 0} \frac{\log(\cos x)}{x} \), we can follow these steps:

### Step 1: Substitute \( x = 0 \)
First, we substitute \( x = 0 \) into the expression:
\[ \cos(0) = 1 \quad \text{and} \quad \log(1) = 0 \]
Thus, the limit becomes:
\[ \frac{\log(\cos(0))}{0} = \frac{0}{0} \]
This is an indeterminate form.

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule. This rule states that if the limit results in an indeterminate form, we can take the derivative of the numerator and the derivative of the denominator.

The limit now becomes:
\[ \lim_{x \to 0} \frac{\frac{d}{dx}[\log(\cos x)]}{\frac{d}{dx}[x]} \]

### Step 3: Differentiate the numerator and denominator
1. **Differentiate the numerator**:
\[ \frac{d}{dx}[\log(\cos x)] = \frac{-\sin x}{\cos x} = -\tan x \]
2. **Differentiate the denominator**:
\[ \frac{d}{dx}[x] = 1 \]

### Step 4: Rewrite the limit
Now we can rewrite the limit as:
\[ \lim_{x \to 0} -\tan x \]

### Step 5: Evaluate the limit
As \( x \to 0 \):
\[ \tan(0) = 0 \]
Thus, the limit becomes:
\[ -\tan(0) = -0 = 0 \]

### Final Result
Therefore, the limit is:
\[ \lim_{x \to 0} \frac{\log(\cos x)}{x} = 0 \]

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