Problem to discuss the differentiability of $f(x)=\sqrt{3x-x^2-2}$
If $f$ is defined on $[a,b]$, then you cannot speak of $$\lim_{h\to 0^-} \frac{f(a+h)-f(a)}{h}$$ because, for $h<0$, the expression $f(a+h)$ is not defined. It is not defined because $a+h<a$ and therefore, $a+h\notin [a,b]$.
Applying this to your case, the limit of $\frac{f(1+h)-f(1)}{h}$, as $h\to 0^-$, does not exist, because for $h<0$, the expression $f(1+h)$ itself is not defined.