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

Options

A. 0
B. a=0
C. a=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: Identify the form of the limit
When we substitute \( x = 0 \) into the expression, we get:
\[ \log(\cos(0)) = \log(1) = 0 \]
Thus, the limit takes the form \( \frac{0}{0} \), which is indeterminate.

### 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 \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), we can take the derivative of the numerator and the derivative of the denominator.

The derivative of the numerator \( \log(\cos x) \) is:
\[ \frac{d}{dx}[\log(\cos x)] = \frac{-\sin x}{\cos x} = -\tan x \]

The derivative of the denominator \( x \) is:
\[ \frac{d}{dx}[x] = 1 \]

### Step 3: Rewrite the limit
Now we can rewrite the limit using L'Hôpital's Rule:
\[ \lim_{x \to 0} \frac{\log(\cos x)}{x} = \lim_{x \to 0} \frac{-\tan x}{1} \]

### Step 4: Evaluate the limit
Now we evaluate the limit as \( x \) approaches 0:
\[ \lim_{x \to 0} -\tan x = -\tan(0) = -0 = 0 \]

### Conclusion
Thus, the final result is:
\[ \lim_{x \to 0} \frac{\log(\cos x)}{x} = 0 \]