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 \( \lim_{x \to \frac{\pi}{2}} \frac{\left\lfloor \frac{x}{3} \right\rfloor}{\ln(\sin x)} \), where \( \left\lfloor . \right\rfloor \) denotes the greatest integer function, we will follow these steps:

### Step 1: Evaluate the numerator
First, we need to evaluate the numerator as \( x \) approaches \( \frac{\pi}{2} \):
\[ \frac{x}{3} \text{ as } x \to \frac{\pi}{2} \implies \frac{\frac{\pi}{2}}{3} = \frac{\pi}{6} \]
Now, we apply the greatest integer function:
\[ \left\lfloor \frac{\pi}{6} \right\rfloor \]
Since \( \frac{\pi}{6} \approx 0.523 \), the greatest integer less than or equal to \( \frac{\pi}{6} \) is \( 0 \):
\[ \left\lfloor \frac{\pi}{6} \right\rfloor = 0 \]

### Step 2: Evaluate the denominator
Next, we evaluate the denominator:
\[ \ln(\sin x) \text{ as } x \to \frac{\pi}{2} \]
We know that:
\[ \sin\left(\frac{\pi}{2}\right) = 1 \implies \ln(1) = 0 \]
Thus, as \( x \) approaches \( \frac{\pi}{2} \), \( \ln(\sin x) \) approaches \( 0 \).

### Step 3: Combine the results
Now we can substitute our results back into the limit:
\[ \lim_{x \to \frac{\pi}{2}} \frac{\left\lfloor \frac{x}{3} \right\rfloor}{\ln(\sin x)} = \lim_{x \to \frac{\pi}{2}} \frac{0}{\ln(\sin x)} \]
Since the numerator is \( 0 \) and the denominator approaches \( 0 \) (but not exactly \( 0 \)), we have:
\[ \frac{0}{\text{something tending to } 0} = 0 \]

### Conclusion
Thus, the value of the limit is:
\[ \boxed{0} \]