Final answer:
The limit of tan^-1(ln(x)) as x approaches 0 does not exist.
Step-by-step explanation:
The given expression is:
lim (x->0) tan^-1(ln(x))
To evaluate this limit, we can start by analyzing the behavior of the natural logarithm function as x approaches 0. As x approaches 0 from the positive side, ln(x) goes to negative infinity. Similarly, as x approaches 0 from the negative side, ln(x) also goes to negative infinity. This means that the expression inside the inverse tangent function approaches negative infinity as x approaches 0.
Next, we consider the behavior of the tangent function as the input approaches negative infinity. The tangent function oscillates between positive and negative infinity as the input approaches negative infinity. Therefore, the overall limit of this expression as x approaches 0 does not exist.