One-sided limits of the floor function at a jump

For $x$ slightly less than $2$, such as $1.9, 1.99, 1.999$, we have $\lfloor x \rfloor = 1$ in every case, since these values lie in the interval $[1,2)$. So the left-hand limit is $\lim_{x \to 2^-} g(x) = 1$.

For $x$ slightly greater than $2$, such as $2.1, 2.01, 2.001$, we have $\lfloor x \rfloor = 2$ in every case, since these values lie in $[2,3)$. So the right-hand limit is $\lim_{x \to 2^+} g(x) = 2$. Since $1 \neq 2$, the one-sided limits disagree, so $\lim_{x \to 2} g(x)$ does not exist.