the value of

lim x tends to infinity cot^-1( log(sinx/[x])) where [.] denotes greatest integer function

To evaluate the limit of cotangent inverse as \( x \) approaches infinity for the expression \( \cot^{-1} \left( \log \left( \frac{\sin x}{[x]} \right) \right) \), we need to break down the components involved in this limit. Let's analyze each part step by step.

Understanding the Components

First, we need to consider the behavior of \( \sin x \) and the greatest integer function \( [x] \) as \( x \) increases without bound.

• Behavior of \( \sin x \): The sine function oscillates between -1 and 1 for all real numbers. Therefore, as \( x \) approaches infinity, \( \sin x \) does not settle at a particular value but continues to oscillate.
• Greatest Integer Function \( [x] \): The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \). As \( x \) increases, \( [x] \) also increases and approaches \( x \) itself.

Analyzing the Logarithmic Expression

Now, let's look at the expression \( \log \left( \frac{\sin x}{[x]} \right) \). Since \( [x] \) grows without bound, we can analyze the fraction:

• As \( x \) becomes very large, \( [x] \) will dominate \( \sin x \) because \( \sin x \) is bounded between -1 and 1.
• This means that \( \frac{\sin x}{[x]} \) will approach 0 as \( x \) approaches infinity.

Consequently, we can say:

\( \frac{\sin x}{[x]} \to 0 \) as \( x \to \infty \).

Evaluating the Logarithm

Next, we take the logarithm of this fraction:

\( \log \left( \frac{\sin x}{[x]} \right) = \log(\sin x) - \log([x]) \).

Since \( \log([x]) \) approaches \( \log(x) \) as \( x \) becomes large, and \( \log(\sin x) \) oscillates (but remains finite), we can conclude:

• As \( x \to \infty \), \( \log([x]) \to \log(x) \to \infty \).
• Thus, \( \log \left( \frac{\sin x}{[x]} \right) \to -\infty \) because \( \log(\sin x) \) is bounded while \( \log([x]) \) grows without bound.

Finding the Limit of Cotangent Inverse

Now we substitute this result back into our original limit:

\( \cot^{-1} \left( \log \left( \frac{\sin x}{[x]} \right) \right) \to \cot^{-1}(-\infty) \).

The cotangent inverse function, \( \cot^{-1}(y) \), approaches \( \pi \) as \( y \) approaches \( -\infty \). Therefore:

\( \lim_{x \to \infty} \cot^{-1} \left( \log \left( \frac{\sin x}{[x]} \right) \right) = \pi \).

Final Result

In conclusion, the value of the limit is:

\( \lim_{x \to \infty} \cot^{-1} \left( \log \left( \frac{\sin x}{[x]} \right) \right) = \pi \).