lim as x goes to 0 from the positive direction of sqrt(x)log(x)

correct me if I'm wrong but the actual limit at x=o does not exist yes?

2. $\lim_{x \to 0^{+}} \sqrt{x} \ln x = \lim_{x \to 0+} \frac{\ln x}{\frac{1}{\sqrt{x}}} = \lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{-1}{2x^{\frac{3}{2}}}} = \lim_{x \to 0^{+}} -2 \sqrt{x} = 0$

3. Thanks a heap, but does the limit at x=0 exist?

4. Does 0 exist?