Is $\lim_{x \to a} \sqrt[K(x)]{L(x)},$ indeterminate, where $\lim_{x \to a} K(x)=\infty, \lim_{x \to a} L(x)=\infty$? Is $\lim_{x \to a} \sqrt[K(x)]{L(x)},$ indeterminate, where $\lim_{x \to a} K(x)=\infty, \lim_{x \to a} L(x)=\infty$?

I do not know how one would show that this is true or otherwise.
It depends.  Let K(x)=L(x)=x, with a infinite. The expression converges to 1. I presume one could make up example where it is indeterminate.
It depends.  Let K(x)=L(x)=x, with a infinite. The expression converges to 1. I presume one could make up example where it is indeterminate.