One-sided limits and existence of a two-sided limit at a piecewise boundary

For the left-hand limit, since $x < 2$, we use $g(x) = 3x+1$. As $x \to 2^-$, this expression approaches $3(2)+1 = 7$, so $\lim_{x\to 2^-} g(x) = 7$.

For the right-hand limit, since $x \ge 2$, we use $g(x) = x^2$. As $x \to 2^+$, this expression approaches $2^2 = 4$, so $\lim_{x\to 2^+} g(x) = 4$.

Since the left-hand limit (7) and the right-hand limit (4) are not equal, the two-sided limit $\lim_{x\to 2} g(x)$ does not exist. This is a case where the one-sided limits disagree, so the function does not settle to a single value as $x$ approaches 2.