Evaluate the limit.lim x → ∞ ln(ln(9x))9x

To evaluate the limit of the function as x approaches infinity, we can use L'Hopital's Rule. This 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 ln(ln(9x))/9x.

First, we need to find the derivative of the numerator and the denominator.

The derivative of ln(ln(9x)) is 1/(ln(9x)) * 1/x = 1/(xln(9x)).

The derivative of 9x is 9.

Now, we substitute these derivatives back into the function and take the limit as x approaches infinity:

lim (x → ∞) [1/(xln(9x))/9] = lim (x → ∞) [1/(9xln(9x))].

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