The value of `lim_(xrarr(pi)/(2))([(x)/(3)])/(ln(sinx))` (where, `[.]` denotes the greatest integer function)

Options

A. does not exist
B. is equal to 1
C. is equal to 0
D. is equal to `-1`

Answer

To solve the limit problem \( \lim_{x \to \frac{\pi}{2}} \frac{\left\lfloor \frac{x}{3} \right\rfloor}{\ln(\sin x)} \), we will follow these steps:

### Step 1: Evaluate the limit of the numerator
First, we need to evaluate the numerator \( \left\lfloor \frac{x}{3} \right\rfloor \) as \( x \) approaches \( \frac{\pi}{2} \).

\[ \frac{\pi}{2} \approx 1.57 \quad \Rightarrow \quad \frac{\pi}{2} \div 3 \approx 0.523 \]

Since \( \left\lfloor 0.523 \right\rfloor = 0 \), we have:

\[ \left\lfloor \frac{x}{3} \right\rfloor \to 0 \quad \text{as } x \to \frac{\pi}{2} \]

### Step 2: Evaluate the limit of the denominator
Next, we evaluate the denominator \( \ln(\sin x) \) as \( x \) approaches \( \frac{\pi}{2} \).

\[ \sin\left(\frac{\pi}{2}\right) = 1 \quad \Rightarrow \quad \ln(1) = 0 \]

Thus, as \( x \to \frac{\pi}{2} \):

\[ \ln(\sin x) \to 0 \]

### Step 3: Form of the limit
Now we have the limit in the form \( \frac{0}{0} \):

\[ \lim_{x \to \frac{\pi}{2}} \frac{\left\lfloor \frac{x}{3} \right\rfloor}{\ln(\sin x)} = \frac{0}{0} \]

### Step 4: Apply L'Hôpital's Rule
Since we have an indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule. However, since the numerator is exactly 0, we can directly evaluate the limit without needing to differentiate.

### Step 5: Evaluate the limit
As we established earlier, since the numerator approaches 0 and the denominator approaches 0, we can conclude:

\[ \lim_{x \to \frac{\pi}{2}} \frac{0}{\ln(\sin x)} = 0 \]

### Conclusion
Thus, the value of the limit is:

\[ \boxed{0} \]