Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.lim x→0 xtan−1(3x)

To find the limit of the function as x approaches 0, we can use 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.

The function we are dealing with is in the form of 0/0 when x approaches 0, which is an indeterminate form. Therefore, we can apply L'Hopital's Rule.

The function is f(x) = x * tan^(-1)(3x). We can rewrite this as f(x) = x / (1/tan(3x)) to clearly see the quotient of two functions.

Now we find the derivative of the numerator and the denominator.

The derivative of x with respect to x is 1.

The derivative of (1/tan(3x)) with respect to x is a bit more complicated. We can use the chain rule and the fact that the derivative of tan(x) is sec^2(x). The derivative is -3/(sec^2(3x) * (3x)^2).

Now we substitute these derivatives back into the limit:

lim (x→0) [1 / (-3/(sec^2(3x) * (3x)^2))].

This simplifies to:

lim (x→0) [-sec^2(3x) * (3x)^2 / 3].

As x approaches 0, this limit is 0.

Therefore, the limit of the original function as x approaches 0 is 0.