The value of `lim_(xrarr(pi)/(2))([(x)/(3)])/(ln(1+cotx))` is equal to (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 given by

\[ \lim_{x \to \frac{\pi}{2}} \frac{\left\lfloor \frac{x}{3} \right\rfloor}{\ln(1 + \cot x)} \]

we will break it down step by step.

### Step 1: Evaluate the numerator

The numerator is \(\left\lfloor \frac{x}{3} \right\rfloor\). As \(x\) approaches \(\frac{\pi}{2}\):

\[ \frac{x}{3} \to \frac{\frac{\pi}{2}}{3} = \frac{\pi}{6} \]

Now, we need to find \(\left\lfloor \frac{\pi}{6} \right\rfloor\). Since \(\frac{\pi}{6} \approx 0.523\), the greatest integer less than or equal to \(\frac{\pi}{6}\) is 0.

Thus,

\[ \left\lfloor \frac{x}{3} \right\rfloor \to 0 \]

### Step 2: Evaluate the denominator

The denominator is \(\ln(1 + \cot x)\). As \(x\) approaches \(\frac{\pi}{2}\):

\[ \cot x = \frac{\cos x}{\sin x} \]

At \(x = \frac{\pi}{2}\), \(\sin x \to 1\) and \(\cos x \to 0\), hence:

\[ \cot x \to 0 \]

Therefore,

\[ 1 + \cot x \to 1 + 0 = 1 \]

Thus,

\[ \ln(1 + \cot x) \to \ln(1) = 0 \]

### Step 3: Analyze the limit

Now we have:

\[ \lim_{x \to \frac{\pi}{2}} \frac{0}{0} \]

This is an indeterminate form. However, since the numerator is exactly 0 and the denominator tends to 0, we can conclude that:

\[ \frac{0}{\text{small positive number}} \to 0 \]

### Conclusion

Thus, the limit evaluates to:

\[ \lim_{x \to \frac{\pi}{2}} \frac{\left\lfloor \frac{x}{3} \right\rfloor}{\ln(1 + \cot x)} = 0 \]

Finally, since we need to find the greatest integer function of this limit, we have:

\[ \left\lfloor 0 \right\rfloor = 0 \]

### Final Answer

The value of the limit is

\[ \boxed{0} \]