The limit $\lim_{x \to 0} \sqrt{x}$ and the existence of limits in simlar cases

Question

If I'm not wrong, $\lim_ {x \to 0} \sqrt{x} = 0$ , even though $\lim_ {x \to 0^{-}} \sqrt{x}$ does not exist.

Then in this following graph of $f(x)$ :

Does $\lim_{x \to 1} f(x)$ exist? Would it still exist if $f(1)$ were not defined?

Answer 1

Since $\sqrt x$ is undefined when $x<0$, it doesn't make sense to talk about $\lim_{x\to0^-}\sqrt x$.

And, yes, $\lim_{x\to1}f(x)=1$, even if $f(1)$ is undefined.