find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not\nlim\n3…

Explanation:

Step1: Analyze the limit of the inner function

As (x\to0^{+}), we know that (\lim_{x\to0^{+}}\ln(x)=-\infty).

Step2: Use the limit property of the arctangent function

We know that (\lim_{u\to-\infty}\tan^{- 1}(u)=-\frac{\pi}{2}). Let (u = \ln(x)), so (\lim_{x\to0^{+}}\tan^{-1}(\ln(x))=\lim_{u\to-\infty}\tan^{-1}(u)=-\frac{\pi}{2}).

Step3: Find the final limit

Using the constant - multiple rule for limits (\lim_{x\to a}k\cdot f(x)=k\cdot\lim_{x\to a}f(x)) (where (k = 3) and (f(x)=\tan^{-1}(\ln(x)))), we have (\lim_{x\to0^{+}}3\tan^{-1}(\ln(x))=3\lim_{x\to0^{+}}\tan^{-1}(\ln(x))). Substituting the result from Step 2, we get (3\times(-\frac{\pi}{2})=-\frac{3\pi}{2}).

Answer:

(-\frac{3\pi}{2})