Does $\lim_{x\to 0^+} \sqrt{x}$ exist or not?

Both $$ \lim_{x\to 0} \sqrt{x} \quad \text{and}\quad \lim_{x\to 0^+}\sqrt{x} $$ exist. In general for a function $f$ with domain $D(f)$, recall the definition of the $$ \lim_{x\to a} f(x) = L. $$ The definition says that this means that: For all $\epsilon >0$ there is a $\delta >0$ such that if $x\in D(f)$ and $0<\lvert x - a \rvert < \delta$ then $\lvert f(x) - L\rvert<\epsilon$. Often we don't write in the requirement that $x$ be in the domain of $f$, but this is a requirement.

Likewise the right hand limit exists.

See this Wikipedia article for more on this: https://en.wikipedia.org/wiki/%28%CE%B5,_%CE%B4%29-definition_of_limit#Precise_statement