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

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

we will follow these steps:

### Step 1: Evaluate the Numerator
First, we need to evaluate the numerator as \( x \) approaches \( \frac{\pi}{2} \).

\[ \frac{x}{3} \text{ as } x \to \frac{\pi}{2} \Rightarrow \frac{\frac{\pi}{2}}{3} = \frac{\pi}{6}. \]

Now, we apply the greatest integer function (denoted by \( \lfloor . \rfloor \)):

\[ \lfloor \frac{\pi}{6} \rfloor. \]

Since \( \frac{\pi}{6} \approx 0.523 \), the greatest integer less than or equal to \( \frac{\pi}{6} \) is \( 0 \):

\[ \lfloor \frac{\pi}{6} \rfloor = 0. \]

### Step 2: Evaluate the Denominator
Next, we evaluate the denominator:

\[ \ln(1 + \cot x) \text{ as } x \to \frac{\pi}{2}. \]

We know that

\[ \cot \left(\frac{\pi}{2}\right) = 0, \]

so we have:

\[ \ln(1 + \cot(\frac{\pi}{2})) = \ln(1 + 0) = \ln(1) = 0. \]

### Step 3: Analyze the Limit
Now we can substitute our findings into the limit:

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

This is an indeterminate form, so we need to analyze it further.

### Step 4: Applying L'Hôpital's Rule
Since we have a \( \frac{0}{0} \) form, we can apply L'Hôpital's Rule. However, since the numerator is \( 0 \) and the denominator approaches \( 0 \), we can conclude that:

\[ \lim_{x \to \frac{\pi}{2}} \frac{0}{\text{something approaching } 0} = 0. \]

### Conclusion
Thus, the limit evaluates to:

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

Finally, since we are dealing with the greatest integer function, the value of the limit is:

\[ \lfloor 0 \rfloor = 0. \]

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

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