`"lt"_(x to infty) (log x)/[[x]]`, where [•] has the usual meaning is

Options

A. 1
B. `-1`
C. 0
D. none

Answer

To solve the limit \( \lim_{x \to \infty} \frac{\log x}{[x]} \), where \([x]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Understand the components of the limit
The function \(\log x\) grows without bound as \(x\) approaches infinity, and \([x]\) (the greatest integer less than or equal to \(x\)) also approaches infinity as \(x\) approaches infinity.

### Step 2: Identify the form of the limit
As \(x\) approaches infinity, both the numerator \(\log x\) and the denominator \([x]\) approach infinity. Thus, we have an indeterminate form of type \(\frac{\infty}{\infty}\).

### Step 3: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that if \(\lim_{x \to c} \frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
provided the limit on the right exists.

### Step 4: Differentiate the numerator and denominator
- The derivative of the numerator \(\log x\) is:
\[ \frac{d}{dx}(\log x) = \frac{1}{x} \]
- The derivative of the denominator \([x]\) is not straightforward because \([x]\) is a step function. However, for large \(x\), \([x] = x\) (the greatest integer function approaches \(x\) as \(x\) becomes very large). Therefore, we can consider the derivative of \(x\):
\[ \frac{d}{dx}([x]) = 1 \quad \text{(for large } x\text{)} \]

### Step 5: Rewrite the limit using derivatives
Now we can rewrite the limit using the derivatives:
\[ \lim_{x \to \infty} \frac{\log x}{[x]} = \lim_{x \to \infty} \frac{\frac{1}{x}}{1} \]

### Step 6: Evaluate the limit
Now we simplify the limit:
\[ \lim_{x \to \infty} \frac{1}{x} = 0 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to \infty} \frac{\log x}{[x]} = 0 \]

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