The value of : `lim_(xrarroo)("log" x)/(x)` is:

Options

A. 1
B. -1
C. 0
D. none of the above

Answer

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

### Step 1: Identify the form of the limit
As \( x \) approaches infinity, \( \log x \) approaches infinity and \( x \) also approaches infinity. Therefore, we have the form \( \frac{\infty}{\infty} \), which is an indeterminate form.

**Hint:** When you encounter an indeterminate form like \( \frac{\infty}{\infty} \), you can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, we can differentiate the numerator and the denominator separately.

- The derivative of the numerator \( \log x \) is \( \frac{1}{x} \).
- The derivative of the denominator \( x \) is \( 1 \).

Thus, we rewrite the limit as:
\[ \lim_{x \to \infty} \frac{\log x}{x} = \lim_{x \to \infty} \frac{\frac{d}{dx}(\log x)}{\frac{d}{dx}(x)} = \lim_{x \to \infty} \frac{\frac{1}{x}}{1} \]

**Hint:** Remember that L'Hôpital's Rule can be applied when you have indeterminate forms.

### Step 3: Simplify the limit
Now, we simplify the limit:
\[ \lim_{x \to \infty} \frac{1}{x} \]
As \( x \) approaches infinity, \( \frac{1}{x} \) approaches \( 0 \).

**Hint:** Consider the behavior of the function as \( x \) becomes very large.

### Step 4: Conclude the result
Thus, we conclude that:
\[ \lim_{x \to \infty} \frac{\log x}{x} = 0 \]

**Final Answer:** The value of the limit is \( 0 \).