Evaluate the limit
\[
\lim _{x \rightarrow \infty} x\left(\frac{\pi}{2}-\frac{1}{x}-\arctan (3 x)\right)
\]

Step 1 ：Given the limit \(\lim _{x \rightarrow \infty} x\left(\frac{\pi}{2}-\frac{1}{x}-\arctan (3 x)\right)\)

Step 2 ：This limit is of the form 0/0 as x approaches infinity, so we can apply L'Hopital's rule. L'Hopital's rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

Step 3 ：First, we need to find the derivative of the numerator and the denominator. The derivative of x is 1. The derivative of the function inside the parentheses can be found using the chain rule and the fact that the derivative of arctan(x) is 1/(1+x^2).

Step 4 ：The derivative of the function is \(x*(-3/(9*x^2 + 1) + x^{-2}) - \arctan(3*x) + \pi/2 - 1/x\)

Step 5 ：After finding the derivatives, we substitute x with infinity and see if the limit exists. If it does, that's our answer. If it doesn't, we may need to apply L'Hopital's rule again or use other methods to find the limit.

Step 6 ：The limit of the derivative as x approaches infinity is 0. This means that the original function approaches a constant value as x approaches infinity.

Step 7 ：Final Answer: \(\boxed{0}\)